^Wi 


i    1    I  I 


^^'^ 


IN  MEMORIAM 
FLORIAN  CAJORl 


INTERMEDIATE  ARITHMETIC 


BY 

BRUCE    M.  WATSON 

SUPERINTENDENT    OF    SCHOOLS,    SPOKANE,    WASH. 

AND 

CHARLES    E.   WHITE 

PRINCIPAL   OF   FRANKLIN   SCHOOL,    SYRACUSE,   N.Y. 


D.    C.    HEATH   &    CO.,    PUBLISHERS 

BOSTON        NEW   YORK         CHICAGO 


Copyright,  1907, 

By  D.  C.  Heath  &  Co. 

1a2 


INTEODUCTION 

The  Intermediate  Arithmetic  is  divided  into  two  parts,  each 
containing  a  full  year's  work.  Throughout  the  book  the  pupil 
is  led  to  see,  in  each  new  topic,  an  extension  and  application  of 
principles  previously  learned.  Fractions  are  treated  as  expres- 
sions of  division.  The  work  in  decimals  is  presented  as  an 
extension  of  the  decimal  scale  of  notation  to  numbers  smaller 
than  one.  Percentage  is  treated  as  an  application  of  decimals 
—  a  familiar  topic  under  a  new  name.  There  is  no  formal  divi- 
sion of  percentage  into  "  cases,"  but  the  pupil  is  led  to  apply 
his  knowledge  of  the  relation  of  product  and  factors  in  deter- 
mining the  process  to  be  employed  in  the  solution  of  each  indi- 
vidual problem.  All  technical  commercial  terms  are  reserved 
for  later  consideration. 

The  work  in  denominate  numbers  is  confined  to  such  prob- 
lems as  people  of  the  present  generation  are  likely  to  meet  in 
their  daily  vocations.  Extended  reductions  and  intricate 
measurements  are  not  required.  Attention  has  been  given  to 
the  development  of  ideas  of  proportion,  the  real  purpose  of 
the  so-called  "  ratio  exercises  "  found  in  some  courses  of  study. 

An  effort  has  been  made  to  shorten  the  course  and  simplify 
the  work,  to  a  reasonable  degree,  by  reducing  each  topic  to  as 
few  cases  as  possible,  and  by  employing  the  simplest  and  most 
generally  applicable  processes. 

The  study  of  arithmetic  should  furnish  the  child  a  means  of 
interpreting  mathematically  the  world  about  him.     It  should 

iii 


iv  INTRODUCTION 

enable  him  to  measure  and  relate  the  facts  of  geography, 
history,  and  science.  It  should  bear  directly  upon  the  vital 
interests  of  the  home  and  family.  Care  has  been  taken  in  the 
selection  of  problems  to  meet  these  requirements,  so  far  as 
possible,  with  due  regard  to  the  mathematical  content  of  the 
exercises,  which  must  always  be  the  first  consideration. 


I 


INTERMEDIATE   ARITHMETIC 


^ 


IIs^TERMEDIATE   ARITHMETIC 

PART   ONE 

NOTATION  AND  NUMERATION 

1.  That  which  tells  how  many  is  number;  e.g.  11,  14  (books), 
25  (cents). 

2.  One  is  a  unit ;  e.g,  1  dollar,  1  house,  one. 

Every  number  is  made  up  of  units.     Three  contains  3  units. 
Twenty  contains  20  units. 

3.  Expressing  numbers  in  figures  or  letters  is  notation ;   e.g, 
7,  29,  VII,  XXIX. 

4.  Expressing  numbers  by  means  of  figures  is  Arabic  notation ; 
e.g.  13,  4728,  23806. 

1,  2,  3,  4,  5,  6,  7,  8,  and  9  are  called  significant  figures,  because 
each  figure  has  a  value. 

The  figure  0,  called  zero,  or  naught,  has  no  value,  but  is  used 
to  give  the  significant  figures  their  proper  places  in  a  number. 

5.  Expressing  numbers  by  means  of  letters  is  Roman  notation : 
e.g.  VIII,  CD,  XCIV. 

3 


ARABIC   NOTATION 
ARABIC   NOTATION 

! 

h  "5 


o 


3 


rt 


o 


Jz;  00  s  H 


<       ^  o  <2 


■C       =        .-       "D 


£     ^     - 


O        TJ       "V        3       -O 
C       =        C        C        O        C 

=      3a>--3a>j=3 


J^        X|-CQII-Sl|-HX|-3 

46    5,  20    9,  315,  087 

This  number  is  read,  four  hundred  sixty-five  billion^  two  huu" 
dred  7iine  million^  three  hundred  fifteen  thousand^  eighty-seven. 

A  comma  (,),  sometimes  called  a  separatrix,  is  used  between 
periods  to  aid  in  reading  numbers. 

7.  Oral 

1.  Name  the  periods  in  this  number.     Name  the  places. 

2.  How  many  periods  are  there  ?     How  many  places  ? 

3.  How  many  places  are  there  in  each  period  ? 

4.  How  does  the  name  of  each  period  compare  with  the 
name  of  its  right-hand  place  ? 

8.  Written 

Express  in  figures:  . 

1.  Two  hundred  thousand,  sixteen. 

2.  Eleven  thousand,  two. 


NUMERATION  5 

3.  Four  million,  six  hundred  eight  thousand,  three  hundred 
seventy-five. 

4.  Twenty-five  thousand,  seven. 

5.  Nineteen  thousand,  seventeen. 

6.  Twenty-seven  million,  six  hundred  fifty. 

7.  Eighty  million,  six  hundred  nine  thousand,  four  hundred 
twenty-eight. 

8.  Six   hundred  twenty  million,  seventeen  thousand,  four 
hundred  seventy-seven. 

9.  Four  hundred  thirty-six  thousand,  fifty-one. 

10.  One  hundred  fifty-seven  million,  six  hundred  eight  thou- 
sand, four  hundred  seventy-seven. 

11.  Three  billion,  fifty-seven  million,  four  hundred  seventeen 
thousand,  sixty. 

12.  Write  a  number  containing  five  places. 

13.  Write  a  number  containing  three  periods.     How  many 
places  does  it  contain  ? 

14.  Write  a  number  containing  three  periods  in  Avhich  the 
thousands'  period  has  no  value. 

NUMERATION 
9.  Naming  the  places  of  figures  and  reading  numbers  is  numer- 
ation. Thus,  to  numerate  43,008,160,  we  should  say  "Units, 
tens,  hundreds,  thousands,  ten-thousands,  hundred-thousands, 
millions,  ten-millions  —  forty-three  million  eight  thousand  one 
hundred  sixty." 

10.    Numerate  the  numbers  below  : 

1.  385  4.         315,129  7.  8,460,000 

2.  1,421  5.      6,785,342  8.      423,000,501 

3.  25,678  6.    35,000,730  9.    8,003,040,631 


KOMAN  NOTATIOIT 


ROMAN  NOTATION 


11.  The  Roman  notation  uses  seven  capital  letters  to  express 
numbers,  as  follows  : 

I  (1),  V  (5),  X  (10),  L  (50),  C  (100),  D  (500),  M  (1000). 

12.  These  letters  are  combined  according  to  the  following 
principles  of  Roman  notation: 

1.  Placing  a  letter  after  one  of  greater  value  adds  its  value  to 
that  of  the  greater, 

2.  Placing  a  letter  before  one  of  greater  value  subtracts  its  value 
from  that  of  the  greater. 

3.  Placing  a  letter  between  two  letters  of  greater  value  subtracts 
its  value  from  their  sum, 

4.  Repeating  a  letter  repeats  its  value, 

5.  Placing  a  bar  over  a  letter  multiplies  the  value  of  the  letter 
hy  1000. 

ILLUSTRATIONS 

13.   1.    X  =  10,   V  =  5,  XV  =  10 +  5  =15.     Which  prin- 
ciple does  this  illustrate  ? 

2.  V  =  5,  I  =  1,  IV  =  5  —  1  =  4.     Which  principle  does 
this  illustrate  ? 

3.  C  =  100,  L  =  50,  X  =  10,  CXL  =  100  +  50  -  10  =  140. 
Which  principle  does  this  illustrate  ? 

4.  X  =  10,  XXX  =  10  +  10  +  10  =  30.     Which  principle 
does  this  illustrate  ? 

5.  D  =  500.    D  =  500000.     Which  principle  ? 

6.  CCCLX  =  360.     Which  principle  ? 

7.  MCM  =  1900.     Which  principle  ? 

8.  MDCLXVI  =  1666.     Which  principle  ? 

9.  Write  a  number  to  illustrate  each  principle. 


ADDITION  7 

10.    Express  in  Roman  notation  all  numbers  from  1  to  100. 

Note.  —  For  many  years  the  Roman  notation  was  the  one  chiefly  used  in 
Europe.  The  ancient  Greeks  also  had  a  system  of  notation  that  employed 
the  letters  of  the  Greek  alphabet.  Both  of  these  systems  were  awkward 
and  were  not  easily  used  in  making  computations.  The  Arabic  system  of 
notation,  now  employed  by  all  the  great  nations  of  the  world,  was  used  first 
in  India,  and  afterward  brought  to  Europe  by  the  Arabs. 

ADDITION 

14.  Addition  is  the  process  of  uniting  two  or  more  numbers  into 
I     one  number  ;  e.g.  2  +  5=7. 

15.  The  numbers  added  are  addends;  e.g,  3  +  10  =  13;  3 
and  10  are  the  addends. 

16.  The  result  of  addition  is  the  sum ;  e.g,  8  books  and  7 
books  are  15  books;  15  is  the  sum. 

17.  The  addends  and  the  sum  are  called  the  terms  of  addition. 

18.  Oral 


Add: 

1.    31 
5 

71 

7 

63 

2' 
8." 

56 

61 
6/ 

46 

8 

34 

7 

26 

« 

19 

9 

9 

71 

Note.  —  In  adding  a  column,  always  look  for  combina- 
tions of  two  or  three  figures  whose  sum  may  be  taken  as 
an  addend.  Thus,  in  finding  the  sum  in  Example  1,  say 
9,  19,  26,  34,  46,  56,  63,  71. 

Read  the  column  downward,  making  similar  combina- 
tions. 


8 

ADDITION 

Add: 

2.    4         3. 

9 

4.    5 

5.    6 

6. 

8 

7.    8 

8.    5 

9.    8 

5 

4 

9 

6 

7 

6 

7 

2 

6 

6 

6 

7 

9 

5 

6 

4 

5 

2 

4 

6 

2 

2 

4 

6 

2 

7 

3 

8 

9 

3 

3 

7 

3 

5 

5 

2 

1 

7 

2 

3 

4 

8 

2 

7 

6 

9 

4 

8 

3 

4 

4 

5 

7 

1 

6 

4 

9 

3 

5 

6 

3 

8 

9 

7 

814^8736 

10.    Add  25  ant?  47. 

25  +  40  =  m.  Say  25,  65,  72. 

65  +  7  =  72.     Ans, 

In  a  similar  way  find  the  sums  indicated  in  exercises  11-25 : 

11.  28  +  26  16.    62  +  28  21.    62  +  24 

12.  42  +  75  17.    57  +  36  22.    65  +  34 

13.  63  +  29  18.    72  +  29  23.    53  +  46 

14.  27  +  38         ^       19.    35  +  26  24.    64  +  44 

15.  45  +  34  20.    44  +  38  25.    73  +  27 

26.  A  drover  bought  8  cows,  5  horses,  and  10  sheep.  How 
many  animals  did  he  buy  ? 

27.  Fred  paid  10  dollars  for  a  goat,  and  12  dollars  for  a  cart 
How  much  did  both  cost  him  ? 

28.  In  a  certain  class  28  pupils  were  present,  5  were  absent 
on  account  of  sickness,  and  4  were  absent  for  other  reasons. 
How  many  pupils  belong  to  the  class? 

29.  A  farmer  sold  25  bushels  of  apples  to  one  man,  10  to 
another,  and  8  to  another.     How  many  bushels  did  he  sell? 


ADDITIOI!^  9 

30.  John  had  40  cents  in  his  bank.  He  added  8  cents  on 
Monday,  and  10  cents  on  Wednesday.  How  much  money  had 
he  then  in  his  bank  ? 

31.  A  man  paid  6  dollars  for  paint,  and  10  dollars  for  labor. 
How  much  did  he  pay  for  both? 

32.  Bought  sheep  for  50  dollars,  turkeys  for  12  dollars,  and 
chickens  for  8  dollars.     How  much  did  they  cost? 

33.  A  man  in  repairing  his  house  paid  65  dollars  for  lumber, 
8  dollars  for  paint,  1  dollar  for  nails,  and  30  dollars  for  labor. 
What  was  the  cost  of  his  repairs? 

34.  How  many  fish  did  Mr.  A  catch  in  four  days  if  he 
caught  12  the  first  day,  8  the  second,  9  the  third,  and  7  the 
fourth  ? 

35.  A  girl  spent  5  cents  for  car  fare,  4  cents  for  pencils,  8  cents 
for  paper,  10  cents  for  ribbon,  15  cents  for  lunch,  and  had  9  cents 
left.     How  much  money  had  she  at  first? 

19.    Written 

Add,  and  test  your  work  hy  adding  downward: 

1. 


28 

2.  639 

3.  1050 

4.   126 

5.  $115.85 

39 

874 

394 

149 

'  327.15 

76 

596 

769 

1260 

495.27 

42 

421 

564 

1004 

160.03 

89 

397 

285 

986  . 

598.09 

I§ 

269 

784 

24 

784.06 

97 

7.  857 

8.  283 

9.  $208.40 

10.  1356.24 

98 

943 

2075 

32.03 

35.09 

79 

268 

298 

26.07 

2.15 

68 

207 

963 

18.94 

30.05 

40 

976 

859 

236.29 

5.16 

87  ' 

888 

876 

28.15 

304.29 

10 


) 

ADDITION 

11.  2673 

12.  837 

13. 

628 

14.  8063 

846 

2964 

4307 

259 

1025 

418 

526 

8264 

92 

3825 

8279 

1287 

837 

842 

428 

428 

642 

29 

4273 

3064 

4983 

561 

746 

42379 

8698 

29 

394 

6507 

2789 

387 

786 

93289 

15.  Three  boys  went  fishing,  and  caught  16  perch,  19  pickerel, 
and  8  black  bass.     How  many  fish  did  they  catch  in  all  ? 

16.  Two  trains  starting  from  the  same  place  ran  two  days  in 
opposite  directions.  One  ran  530  miles  the  first  day  and  525 
miles  the  second,  while  the  other  ran  492  miles  the  first  day 
and  510  miles  the  second.  How  far  apart  were  they  at  the  end 
of  the  two  days?     (Illustrate  by  a  picture.) 

17.  A  man  bought  coal  for  $5.60,  wood  for  $3.45,  and  a 
stove  for  $45.     What  was  the  whole  cost? 

18.  There  are  112  bushels  of  wheat  in  one  bin,  175  in  an- 
other, and  234  in  the  third.     How  many  bushels  in  all  ? 

19.  There  are  218  pages  in  my  reader,  245  in  my  arithmetic, 
195  in  my  geography,  and  189  in  my  language  book.  How 
many  pages  in  the  four  books  ? 

20.  The  Bum  of  all  the  sides  of  a  figure  is  its  perimeter. 

1.  Find  the  perimeter  of  a  figure  whose  sides  are  39  inches, 
45  inches,  28  inches,  b^  inches,  75  inches,  and  17  inches. 

2.  What  is  the  perimeter  of  a  seven-sided  piece  of  land 
whose  sides  are  209  feet,  683  feet,  129  feet,  463  feet,  928  feet, 
93  feet,  and  290  feet  ?     (Illustrate.) 


SUBTRACTION  11 

^^  SUBTRACTION 

21.  Subtraction  is  the  process  of  finding  the  difference  between 
two  numbers;  e.g.  21— 1  =  14:;  13  cents  —  5  cents  =  8  cents. 

22.  The  number  from  which  we  subtract  is  the  minuend.  The 
number  subtracted  is  the  subtrahend.  The  result  of  subtraction 
is  the  difference  or  remainder. 

The  difference  is  always  the  number  that  must  be  added  to 
the  subtrahend  to  obtain  the  minuend ;  e.g.  17  —  9  =  8.  17  is 
the  minuend,  9  is  the  subtrahend,  and  8  is  the  difference  or 
remainder. 

23.  77ie  minuend,  subtrahend,  and  remainder  are  called  the 
terms  of  subtraction. 

24.  Oral 

From  83  take  57. 

83  -  50  =  33.  Say  83,  33,  26. 

33  -  7  =  26.     Ans. 

In  a  similar  way  find  the  differences  indicated  in  exercises  1-15. 


1.    38-19 

6. 

72-26 

11. 

63-25 

2.   27-18 

7. 

66-37 

12.. 

,48-19 

3.   42-15 

8. 

92-48 

13. 

51-27 

4.    61-22 

9. 

87-39 

14. 

84-47 

5.    81-36 

10. 

42-29 

15. 

75-39 

16.  Frank  lives  12  blocks  from  school,  and  Henry  5  blocks 
In  the  same  direction.  Their  homes  are  how  many  blocks 
apart  ?  How  many  blocks  apart  would  they  be  if  Henry  lived 
5  blocks  from  school  in  the  opposite  direction  ?     (Illustrate.) 

17.  Mary  added  two  numbers,  and  the  sum  was  28.  One 
of  the  numbers  was  16.     What  was  the  other  ? 

18.  20  is  how  much  more  than  11? 


12  .  SUBTRACTION 

19.  Tom  has  20  marbles,  and  Edward  11.  Tom  has  how 
many  more  than  Edward  ? 

20.  If  you  pay  15  cents  toward  the  purchase  of  a  slate  cost- 
ing 20  cents,  how  much  do  you  still  owe  ? 

21.  Nell  had  21  chickens,  but  a  dog  killed  10  of  them. 
How  many  were  left? 

22.  Lucy  is  20  years  old,  and  her  sister  is  6  years  younger. 
How  old  is  her  sister? 

23.  John  had  1 20.  He  spent  1 10  for  a  coat  and  1 3  for  a 
hat.     How  much  had  he  left  ? 

24.  A  pole  was  19  feet  long.  I  cut  off  4  feet  at  one  time 
and  6  feet  at  another.     How  many  feet  were  left  ? 

25.  What  number  must  be  subtracted  from  43  to  leave  25  ? 

26.  John's  heart  beat  78  times  a  minute  when  he  was  well, 
but  130  times  a  minute  during  a  severe  illness.  How  much 
faster  did  the  heart  beat  during  illness  than  in  health? 

25.    Written 

From  58500  take  26937. 

58500  =  50000  +  7000  + 1400  +  90  +  10 
26937  =  20000  +  6000+  900  +  30+  7 
31563  =  30000  +  1000+   500  +  60+   3 

In  finding  the  difference,  we  write  only  this : 
58500 
26937 

31563  and  say,  7  from  10  =  3,  3  from  9  =  6,  9  from 
14  =  5,  6  from  7  =  1,  2  from  5  =  3. 

To  test  the  work,  add  the  subtrahend  and  remainder.  If  the  minu- 
end is  obtained,  the  work  is  correct.  Do  not  write  the  numbers  again,  but 
make  the  test  with  the  numbers  as  they  stand. 

In  what  other  way  may  we  test  subtraction? 


SUBTRACTION 


13 


Subtract  and  test  results 


1. 

2819 
674 

5. 

2763 

1289 

9. 

92874 
11392 

13. 

38264 
29842 

17. 

12.15 
1.12 

21. 

134.28 
24.28 

2. 


10. 


14. 


82'03 
1276 

37284 

9287 

94210 

8206 

19327 
8291 


18.   $35.28 
17.05 


22. 


.21 
27.13 


3. 
7. 

4295 
597 

36801 
18463 

11. 

42840 
38706 

15. 

92593 

87246 

19. 

$25.18 
1.15 

23. 

$17.80 
16.75 

7306 
1807 


8. 

18003 
921 

12. 

98301 
26942 

16. 

27075 
18092 

20. 

$36.51 
16.82 

24. 

$75.00 
24.32 

25.  From  seventeen  thousand  sixteen,  take  nine  thousand 
four  hundred  eighty-seven. 

26.  From  seventy-two  thousand  three  hundred  eleven,  take 
forty-six  thousand  nine  hundred  sixty-one. 

27.  Take  eight  thousand  four,  from  thirty  thousand! 


REVIEW  AND  PRACTICE 
26.    Oral 

1.  Read  359,016,007,138 ;  $3,894,760.15;  1,010,101. 

2.  ReadCLIX;  DCCXXXVI;  MCMXIIL 

3.  Numerate  3,057,608. 

4.  Jennie  bought  a  skein  of  Shetland  floss  for  10  cents,  3 
skeins  of  embroidery  silk  for  12  cents,  and  a  pair  of  knitting 
needles  for  10  cents.  How  much  change  should  she  receive 
from  a  50-cent  piece? 


14  REVIEW  AND  PRACTICE 

5.  The  sum  of  3  numbers  is  100.    Two  of  them  are  29  and 

37.     What  is  the  other? 

6.  Albert  has  earned  15  cents,  25  cents,  and  17  cents  toward 
a  pair  of  gloves  that  cost  f  1.  How  much  more  money  must 
he  obtain  in  order  to  pay  for  the  gloves  ? 

7.  Helen,  Howard,  and  Henry  wanted  a  canoe  that  cost  $47. 
They  obtained  f  19  by  renting  their  row-boat,  their  mother  con- 
tributed $12,  and  their  father  the  remainder  of  the  price.  How 
much  did  their  father  give  ? 

8.  Edith  made  some  purchases,  gave  the  clerk  a  dollar,  and 
received  in  change  three  cents,  one  nickel,  two  dimes,  and  a 
quarter.     What  was  the  amount  of  her  purchases? 

27.    Written 

1.  A  farmer  having  456  bushels  of  corn  sold  84  bushels  to 
one  man  and  135  bushels  to  another.  How  many  bushels  did 
he  have  left  ? 

2.  A  man  started  to  walk  112  miles  in  three  days.  He 
walked  32  miles  the  first  day,  and  41  miles  the  second.  How 
far  must  he  walk  the  third  day  to  complete  the  journey  ? 

3.  I  bought  a  cow  for  $42,  another  for  $48,  and  a  third  for 
$56.     For  how  much  should  I  sell  them  to  gain  $28? 

4.  A  lady  bought  sugar  for  65  cents,  tea  for  55  cents,  molas- 
ses for  72  cents,  butter  for  84  cents,  starch  for  25  cents,  and 
gave  in  payment  a  five-dollar  bill.  How  much  change  should 
she  receive  ? 

5.  The  distance  by  rail  from  Galveston  to  San  Antonio  is 
572  miles,  from  San  Antonio  to  Tucson  932  miles,  and  from 
Tucson  to  Los  Angeles  501  miles.  What  is  the  distance  by 
rail  from  Galveston  to  Los  Angeles  ? 


REVIEW  AND  PRACTICE  15 

6.  Two  vessels  start  from  points  850  miles  apart  and  sail 
toward  each  other.  How  far  apart  are  they  when  one  has 
sailed  246  miles  and  the  other  352  miles?     (Illustrate.) 

7.  A  man  sold  one  horse  for  $145  and  another  for  $182. 
On  the  first  he  gained  1 23,  and  on  the  second  1 36.  What  was 
the  cost  of  both? 

8.  A  boy  bought  apples  for  $A5  and  pears  for  $.62,  and 
sold  them  all  for  $1.50.     What  was  his  profit? 

9.  John  sold  62  newspapers,  Frank  48,  and  Henry  27  less 
than  both  of  them.     How  many  did  Henry  sell  ? 

10.  A  grocer  sold  butter  for  $45  and  cheese  for  $62.  On  the 
butter  he  lost  $6  and  on  the  cheese  he  gained  $14.  What  was 
the  cost  of  both? 

11.  A  farmer  bought  a  barrel  of  flour  for  $6.35,  sugar  for 
$2.15,  coffee  for  $1.46,  tea  for  $1.20,  and  gave  in  payment 
$3.15  worth  of  butter  and  the  remainder  in  cash.  What  did 
he  pay  in  money? 

12.  The  sum  of  52  and  64  is  how  much  greater  than  the 
difference  between  124  and  69? 

13.  From  a  flock  of  320  sheep  76  were  sold  at  one  time  and 
112  at  another.     How  many  remained  ? 

14.  A  man  bought  148  bushels  of  potatoes  from  A,  216 
bushels  from  B,  183  bushels  from  C,  and  afterwards  sold  all 
but  137  bushels.     How  many  bushels  did  he  sell? 

15.  The  sum  of  three  numbers  is  342.  Two  of  the  numbers 
are  84  and  96.     What  is  the  third  number? 

16.  John's  father  gave  him  $2.25,  and  his  uncle  gave  him 
$1.40.  He  earned  enough  besides  so  that  he  bought,  with  the 
whole,  a  suit  of  clothes  for  $8.     How  much  did  he  earn? 


16  REVIEW  AND  PRACTICE 

17.  Claude  took  987  steps  in  coming  to  school,  Francis  865, 
and  Alice  398  less  than  the  number  taken  by  both  the  boys. 
How  many  steps  did  all  three  take? 

18.  A  ship  loaded  with  iron  sailed  from  Cleveland  to  a 
port  332  miles  west  of  that  city.  A  car  loaded  with  machinery 
at  Cleveland  was  taken  to  a  city  619  miles  east  of  Cleveland. 
How  far  apart  were  the  ship  and  the  car  when  each  had  reached 
the  end  of  its  trip?     (Illustrate.) 

19.  The  first  Thanksgiving  was  in  1621  and  the  day  has  been 
observed  every  year  since.  How  many  times  has  the  day  been 
observed? 

20.  A  father  and  his  three  sons  earned  $2461  in  a  year. 

The  first  son  earned  f  676,  the  second  $456,  and  the  father 
$1080.     How  much  did  the  third  son  earn? 

21.  A  train  started  from  Chicago  with  324  passengers.  On 
the  way  to  St.  Paul  185  passengers  left  the  train,  and  149  came 
aboard.  How  many  passengers  were  on  the  train  when  it 
reached  St.  Paul? 

22.  A  retail  grocer  bought  at  a  wholesale  grocery  three  bar- 
rels of  apples  for  $4.50,  a  box  of  lemons  for  $2,70,  and  three 
barrels  of  flour  for  $  12. 30.  He  handed  the  wholesale  grocer  one 
gold  piece  and  received  50  cents  in  change.  What  was  the 
value  of  the  gold  piece  ? 

23.  During  one  week,  a  man  put  into  the  bank  $687,  drew 
out  $489,  put  in  $348,  drew  oat  $298,  and  then  had  $1386  left 
in  the  bank.     How  much  had  he  in  the  bank  at  first? 

24.  A  farmer  having  215  acres  of  land,  used  21  acres  for 
corn,  36  for  oats,  29  for  barley,  18  for  potatoes,  52  for  meadow, 
and  the  rest  for  pasture.  How  many  acres  were  used  for 
pasture  ? 


MULTIPLICATION 


17 


MULTIPLICATION 

28.  Multiplication  is  taking  one  number  as  many  times  as  there 
are  units  in  another ;  e.g.  6  times  9  are  54. 

29.  The  number  multiplied  is  the  multiplicand  ;  the  number 
by  which  we  multiply  is  the  multiplier  ;  the  result  of  multiplica- 
tion is  the  product;  e.g.  12  times  20  are  240.  20  is  the  multi- 
plicand^ 12  is  the  multiplier^  and  240  is  the  product.  20  and 
12  are  factors  of  240. 

30.  The  multiplier^  multiplicand^  and  product  are  called  the 
terms  of  multiplication. 

31.  The  multiplier  and  multiplicand  are  factors  of  the  product. 
The  product  is  the  same  in  whatever  order  the  factors  are 
taken;  e.g.  6  times  7  are  42,  and  7  times  6  are  42;  3x5x4 
are  60  and  4x3x5  are  60. 


32. 


ORAL   EXERCISES 
(Reviewing  work  of  primary  arithmetic) 


3 

4 

5 

2 

6 

1 

8 

0 

7 

12 

10 

11 

9 

7 

12 

9 

0 

11 

5 

10 

4 

1 

3 

6 

2 

8 

1.  Multiply  every  number  in  the  upper  line  by  each  number 
in  the  lower  line. 

2.  How  do  we  multiply  a  number  by  10  ?    By  100  ?    By  1000  ? 

3.  Multiply    7  by  10  ;  by  100 ;  by  1000. 

4.  Multiply  34  by  10  ;  by  100 ;  by  1000. 

5.  Multiply  11  by    3 ;   by    30;  by  300. 

6.  Multiply  12  by    7  ;  by    70 ;  by  7000. 


18  MULTIPLICATION 

7.  The  product  is  a  multiple  of  the  multiplicand.     Of  what 
other  number  is  it  a  multiple  ? 

8.  Of  what  number  is  33  a  multiple? 

9.  Name  5  divisors  of  24. 

10.  Multiply  23  by  12. 

11.  Multiply  23  by  10,  also  by  2.     Add  the  two  products. 
How  does  this  result  compare  with  the  answer  to  question  10? 

12.  Multiply  30  by  5,  then  by  20.     Add  the  two  products. 
The  sum  is  how  many  times  30  ? 

13.  90  times  48,  added  to  6  times  48,  are  how  many  times 
48? 

14.  7  times  786,  40  times  786,  and  3  times  786,  added,  make 
how  many  times  786? 

Q-ive  the  products  at  sight : 

15.  86  16.   307        17.   315        18.   73  19.   32 

7  5  10  100  20 


20.  14 

21.  28 

22.  25 

23.  16 

24.  28 

30 

200 

300 

700 

1000 

25.  75 

26.  103 

27.  212 

28.  31 

29.  205 

2000 

40 

50 

400 

600 

30.  Edward  feeds  his  horse  11  quarts  of  grain  per  day  and 
his  chickens  3  quarts.  He  feeds  his  horse  how  much  more  in 
30  days  than  he  feeds  his  chickens  in  the  same  time  ? 

31.  Find  the  cost  of  50  five-cent  postage  stamps. 

32.  11  and  11  are  the  factors  of  what  number  ? 

33.  Of  what  number  are  3,  7,  and  5  the  factors  ? 

34.  What  must  be  paid  for  a  dozen  junior  baseballs  at  70 
cents  apiece  ? 


MULTIPLICATION 


19 


33.    Written 
Multiply  1283  by  967. 
1283 


967 
8981 
7698 
11547 
1240661 


We  multiply  1283  by  7,  60,  and  900,  and  then  add  the 
results,  which  are  called  partial  products.  The  sum  of 
the  partial  products  is  the  product  required.  We  omit 
ciphers  at  the  right  of  the  partial  products  after  the  first. 

Read  each  partial  product  as  if  the  ciphers  were  ex- 
pressed. 


Multiply  $34.79  by  806. 


$34.79 

806 

20874 

27832 

128040.74 

1.  324x24 

2.  296x39 

3.  387x45 

4.  263x56 

5.  892x63 

6.  728x75 

7.  398x84 

8.  987x98 

9.  516x31 

10.  798x43 

11.  896x79 

12.  598x36 

13.  287x49 

14.  799x99 


Observe  that  the  right-hand  figure  of  each  partial 
product  is  written  directly  under  the  figure  by  which 
we  multiply  to  obtain  it.  Cents  in  either  factor  give 
cents  in  the  product. 


15.  296x28 

16.  694x39 

17.  206x54 

18.  128.15x28 

19.  134.98x27 

20.  119.84x46 

21.  $7.85x124 

22.  128.75x15 

23.  $36.91x45 

24.  $126.93x87 


25. 
26. 
27. 
28. 


$.17.85x48 
$19.63x49 
$75.10x97 
$16.35x764 


29.  $280.52x236 

30.  $356.04x328 

31.  $987.62x475 

32.  $396.41x641 

33.  $806.04x879 

34.  238x307 

35.  5126x208 

36.  934x9000 

37.  1027x2005 

38.  386x1080 

39.  527x2300 

40.  4008x7003 

41.  $29.05x108 

42.  4040x8356 


20  MULTIPLICATION 

34.  PROBLEMS 

1.  a.  What  is  the  perimeter  of  a  square  farm  whose  side  is 
309  rods  ?     5.  What  is  its  area  ? 

2.  a.  If  it  costs  14.78  a  day  to  support  a  certain  family, 
how  much  does  it  cost  for  a  month  of  31  days  ?  h.  How  much 
does  it  cost  for  a  month  of  28  days  ?  e.  How  much  does  it 
cost  for  a  year  ? 

3.  There  are  2000  pounds  in  one  ton.  How  many  pounds 
are  there  in  496  tons ?     (Solve  it  in  the  shortest  way.) 

4.  There  are  24  hours  in  a  day,  60  minutes  in  an  hour,  and 
60  seconds  in  a  minute,  a.  What  is  the  number  of  seconds  in 
a  day  ?     h.  In  a  week  ? 

5.  a.  How  many  minutes  are  thef^  in  the  month  of  May? 
h.  In  the  month  of  February  ?     c.  In  the  month  of  September  ? 

6.  Alice  hemmed  4  dozen  handkerchiefs,  each  12  inches 
square.     How  many  inches  of  hem  did  she  make  ? 

7.  A  manufacturer  put  up  13  tons  of  cereal  in  sample 
packages,  each  containing  1  ounce.  How  many  packages  did 
he  make  ? 

8.  25,079  crates  of  strawberries,  each  containing  36  quarts, 
were  shipped  from  the  city  of  Oswego  in  one  season.  What 
were  they  worth  at  8  cents  a  quart  ? 

9.  There  are  12  things  in  a  dozen,  12  dozen  in  a  gross,  and 
12  gross  in  a  great  gross.  How  many  pens  in  a  case  containing 
307  great  gross  ? 

10.    What  is  the  value  of  347  barrels  of  shredded  cocoanut, 
each  containing  105  pounds,  at  15  cents  a  pound  ? 


DIVISION  21 


DIVISION 


35.  Division  is  the  process  of  finding  one  of  two  factors  when 
the  other  factor  and  the  product  are  given;  e.g.  35  is  the  prod- 
uct of  5  and  7.  When  35  and  5  are  given,  we  divide  35  by 
5  to  obtain  7 ;  when  35  and  7  are  given,  we  divide  35  by  7  to 
obtain  5. 

36.  The  number  divided  is  the  dividend  ;  the  number  by  which 
we  divide  is  the  divisor,  and  the  result  of  division  is  the  quotient ; 
e.g.  42  -^  7  =  6.  42  is  the  dividend,  7  is  the  divisor,  and  6  is 
the  quotient. 

When  the  divisor  is  not  exactly  contained  in  the  dividend, 
the  part  of  the  dividertd  that  is  left  is  called  the  remainder ;  e.g. 
59  -T-  8  =  7  quotient  and  3  remainder. 

37.  The  dividend,  divisor,  and  quotient  are  called  the  terms  of 
division. 

38.  Oral 

1.  6  oranges  at  3  cents  apiece  cost  18  cents.  Which  of  these 
numbers  is  a  product  ?  Which  are  factors  ?  When  18  and  6 
are  given,  how  can  3  be  found  ?  When  18  and  3  are  given, 
how  can  6  be  found  ?  When  3  and  6  are  given,  by  what  opera- 
tion can  18  be  found  ? 

2.  The  area  of  a  page  of  Henry's  book  is  35  square  inches. 
If  the  length  is  7  inches,  what  must  be  the  width?  If  the 
width  is  5  inches,  what  must  be  the  length  ?  35  is  which  term 
in  division  ?     35  is  what  of  5  and  7  ? 

3.  There  are  9  square  feet  in  1  square  yard.  How  many 
square  yards  are  there  in  108  square  feet  ?  108  is  the  product 
of  9  and  what  other  number  ?  108  is  which  term  in  division  ? 
108  is  what  of  9  and  12  ? 


22  DIVISION 

4.  Fred  paid  54  cents  for  some  sugar.  The  number  of  cents 
that  a  pound  cost  is  one  factor  of  54.  What  is  the  other  fac- 
tor ?  What,  besides  54  cents,  must  be  given  in  order  that  we 
may  find  the  cost  of  one  pound  ?  What,  besides  54  cents,  must 
be  given  in  order  that  we  may  find  the  number  of  pounds  that 
Fred  bought  ? 

5.  What  is 

a.    The  number  of  feet  in  132  inches  ? 
h.    The  number  of  pecks  in  72  quarts  ? 

c.  The  cost  of  a  month's  rent  at  $120  a  year  ? 

d.  The  number  of  weeks  in  77  days  ? 

e.  The  price  of  a  lawn  mower,  when  15  lawn  mowers  cost 
1150? 

6.  The  first  number  in  each  line  below  is  a  factor  of  every 
other  number  in  the  line.  Find  the  factor  not  given  of  each 
number; 

15;     75;  355;  525;     405 

84;  217;  280;  763;  497 
110;  44;  121;  880;  2211 
819;   945;   189;   360;     963 

64;  328;   176;   728;     960 

7.  In  the  following  statements  tell  which  numbers  are  fac- 
tors and  which  are  products : 

a.  There  are peaches  in  7  baskets  if  each  basket  con- 
tains 12  peaches. 

h.    12  quarts  of  berries  cost  96  cents. 

c.  Jerome's  wages  for  9  weeks  at  f a  week  amounted  to 

27  dollars. 

d.  6  fountain  pens  at  $3  apiece  cost  $ . 

e.  il  will  pay  20  car  fares  at apiece. 

f.  72  cents  will  buy  12  pounds  of  sugar  at cents  a  pound. 


a. 

5 

h. 

7 

c. 

11 

d. 

9, 

e. 

8 

DIVISION 


28 


39.    Written 


1. '  Divide 
3125 
317)990625 
951 
396 
317 
792 
634 


1585 
1585 


990625  by  317. 

317  is  not  contained  in  9  or  99,  but  it  is  contained  in 
990  three  times.  This  is  3  thousand  because  990  is  thou- 
sands. The  remainder  is  39  thousand.  Put  600  with  it, 
and  317  is  contained  in  396  hundred  1  hundred  times,  with 
a  remainder  of  79  hundred.  The  remaining  figures  of  the 
quotient  are  obtained  in  a  similar  manner.  When  the 
second  figure  (from  the  left)  of  the  divisor  is  1,  2,  or  3,  it 
is  well  to  use  the  first  figure  for  a  "  guide  figure  "  in  ob- 
taining each  quotient  figure.  Thus,  in  this  example,  we 
say,  3  in  9  =  ?    3  in  3  =  ?    3  in  7  =  ?    3  in  15  =  ? 

Note.  —  Be  careful  to  write  the  first  figure  in  the  quo- 
tient directly  above  the  right-hand  figure  of  the  part  of 
the  dividend  used  in  obtaining  the  first  quotient  figure. 


2.    Divide  10192  by  49. 

208 


49)10192 
98 
392 
892 


In  finding  the  second  quotient  figure,  we 
see  that  49  is  not  contained  in  39 ;  so  we 
place  0  in  tens'  place  in  the  quotient  and 
bring  down  2  units.  49  is  almost  50.  We 
may  therefore  take  5  for  a  guide  figure. 


3.    Divide  8531  by  672 

12|U 

Quotient 

672)8531 

672 

1811 

1344 

In  this  example  the  dividend  does  not 
contain  the  divisor  an  exact  number  of 
times,  hence  there  is  a  remainder.  The  re- 
mainder may  be  written  over  the  divisor  as 
a  part  of  the  quotient ;  thus,  12|f|  quotient. 


Note.  —  In  dividing  by  any  number  not 
larger  than    12,    short    division  should   be 


467  Remainder 

used.  That  is,  no  work  should  be  written  except  the  dividend,  divisor,  and 
quotient.  In  such  examples  the  quotient  may  be  written  either  above  or 
below  the  dividend  according  to  convenience.  If  the  dividend  contains 
cents,  and  the  divisor  is  a  whole  number,  the  quotient  also  contains  cents. 


24  DIVISION 

Solve  examples  4-68,  and  test  your  work  hy  multiplying  the 
quotient  hy  the  divisor  and  adding  the  remainder^  if  there  is  one, 
to  obtain  the  dividend : 


4. 

4503  -  3 

26. 

28,692  -  9 

48. 

130,052  -  2 

5. 

2045  -^  5 

27. 

333,333  ^  11 

49. 

168,754  -  9 

6. 

2835  ^  7 

28. 

35,621  ^  7 

50. 

385,980  -  5 

7. 

4986  -^  9 

29. 

42,963  ^  6 

51. 

769,520  -V-  7 

8. 

2009  -^  7 

30. 

50,725  -r-  3 

52. 

387,052  -  10 

9. 

3504  ^  8 

31. 

82,956  ^  10 

53. 

943,769  -r- 12 

10. 

.6  193a. 

32. 

93,043  ^  7 

54. 

748,136  -  4 

11. 

^_±a 

33. 

65,407  -f-  5 

55. 

7,688  -  31 

12. 

iLa840 

34. 

39,842  -  9 

56. 

12,978  -  42 

13. 

12  3  M. 

35. 

27,392  -  8 

57. 

31,509  -f-  81 

14. 

5  8  3  4.5. 

36. 

63,594  -  9 

58. 

40,948  -^  58 

15. 

11^ 

37. 

31,493  -^  6 

59. 

68,476  -h  68 

16. 

lA&M. 

38. 

25,324  -  5 

60. 

168,665  -H  427 

17. 

2  9  31A 

39. 

28,764  -f-  4 

61. 

190,855  -^  931 

18. 

^|ij6. 

40. 

36,099  H-  9 

62. 

293,004  -  801 

19. 

30,005  ^ 

5 

41. 

14,412  -  12 

63. 

129,324  -  756 

20. 

288,012  - 

12 

42. 

36,930  -f-  11 

64. 

3,247,654^79 

21. 

300,010  ^ 

10 

43. 

24,003  ^  6 

65. 

294,490  =  98  X? 

22. 

99,011  -^ 

11 

44. 

30,502  -  8 

66. 

503x7=637,804 

23. 

33,264  - 

11 

45. 

29,333  -  11 

67. 

58,487  has  what 

24. 

29,280  - 

12 

46. 

675,262  -  5 

factor  besides  143  ? 

25. 

36,550  -^ 

10 

47. 

349,872  -  8 

68. 

?x  215  =  66,220 

REVIEW   AND   PRACTICE  26 

REVIEW   AND  PRACTICE 
40.    Oral 

1.  A  farmer  exchanged  12  barrels  of  apples  at  |3  a  barrel 
for  coal  at  i  4  a  ton.     How  many  tons  of  coal  did  he  receive  ? 

2.  In  what  time  will  a  boy  earn  as  much  at  f  3  a  week,  as  a 
man  earns  in  6  weeks  at  $8  a  week  ? 

3.  Nell  is  3  years  old  and  Will  5.  Their  sister's  age  is  twice 
the  sum  of  their  ages.     What  is  the  sister's  age  ? 

4.  How  many  gallons  of  milk  will  a  family  use  in  the  month 
of  June  if  they  use  2  quarts  a  day  ? 

5.  Frank  rides  6  miles  an  hour  and  Albert  9.  a.  How  far 
apart  will  they  be  in  6  hours  if  they  start  at  the  same  time  and 
place  and  ride  in  the  same  direction  ?  h.  If  they  ride  in  op- 
posite directions  ? 

6.  Grace  bought  2  dozen  lemons.  She  used  |  of  them  for 
lemonade  and  gave  away  6.     How  many  remained? 

7.  Helen  bought  -^  of  a  yard  of  cambric.  She  used  half  a 
yard  in  her  dress  and  wasted  ^  of  a  yard  in  cutting.  How 
much  was  left  ? 

8.  A  man  owed  $96.  He  made  5  payments  of  1 12  each. 
How  much  did  he  then  owe  ? 

9.  Luther  has  %  6  and  Leon  3  times  as  much.  How  much 
have  both? 

10.  118  plus  112  is  how  much  less  than  4  times  $12? 

11.  The  product  is  125  and  one  of  the  factors  is  5.     What 
is  the  other  factor  ? 

12.  The  divisor  is  11,  the  quotient  12,  and  the  remainder  9. 
What  is  the  dividend  ? 

13.  The  dividend  is  85,  the  quotient  9,  and  the  remainder  4. 
What  is  the  divisor  ? 


26  REVIEW  AND  PRACTICE 

14.  How  many  days  will  a  12-gallon  keg  of  water  last  24 
shipwrecked  men  if  each  man  drinks  a  pint  a  day  ? 

15.  If    Frances  can   knit    21  stitches  a  minute,  how  many 
stitches  can  she  knit  in  half  an  hour  ? 

16.  The  product  of  20  and  16  is  how  much  less  than  20  times 
20? 

41.    Written 

1.  How  many  tons  of  coal  at  f  5  a  ton  will  pay  for  15  tons  of 
hay  at  $11  a  ton? 

2.  A  man  started  on  a  journey  of  724  miles.  After  he  had 
traveled  12  hours  at  the  rate  of  32  miles  an  hour,  how  far  was 
he  from  his  journey's  end? 

3.  A  farmer  bought  six  sacks  of  flour  at  §1.25  a  sack,  25 
pounds  of  sugar  at  6  cents  a  pound,  and  two  pounds  of  coffee  at 
30  cents  a  pound.  He  paid  for  it  in  butter  at  24  cents  a  pound. 
How  many  pounds  of  butter  were  there  ? 

4.  A  man  having  1738.58  in  a  bank  drew  out  1132.75  at 
one  time,  f  175.50  at  another,  and  $216  at  another.  How  much 
money  then  remained  in  the  bank  ? 

5.  If  the  divisor  is  38,  the  quotient  209,  and  the  remainder 
23,  what  is  the  dividend? 

6.  The  product  of  three  numbers  is  1260  and  two  of  them 
are  12  and  7.     What  is  the  third? 

7.  A  grocer  buys  88  gallons  of  molasses  at  $.56  a  gallon. 
For  what  price  per  gallon  must  he  sell  it  in  order  to  gain 
112.32? 

8.  The  dividend  is  1821,  the  quotient  32,  and  the  remainder 
29.     What  is  the  divisor? 

9.  Two  trains  start  at  the  same  time  from  points  1216  miles 
apart  and  travel  toward  each  other,  one  at  the  rate  of  35  miles 


REVIEW  AND  PRACTICE  27 

an  hour,  the  other  at  the  rate  of  41  miles  an  hour.     In  how- 
many  hours  will  they  meet? 

10.  A  lady  bought  12  yards  of  dress  goods  at  $1.75  a  yard, 
8  yards  of  silesia  at  f.25  a  yard,  2  pairs  of  gloves  at  $1.45  a 
pair,  6  handkerchiefs  at  8.25  apiece,  and  3  yards  of  table  linen 
at  1.95  a  yard.     She  paid  118.75.     What  did  she  still  owe? 

11.  How  many  pounds  of  cheese  at  14  cents  a  pound  will 
pay  for  3  barrels  of  flour  at  $4.20  a  barrel? 

12.  At  what  rate  per  hour  must  a  train  run  to  go  as  far  in  9 
hours  as  another  train  running  27  miles  an  hour  can  go  in  12 
hours? 

13.  A  farmer  paid  $1125  for  cows,  horses,  and  farming  tools, 
and  8  times  as  much  for  a  farm  of  125  acres.  What  was  the 
price  per  acre? 

14.  A  farmer  had  46  acres  of  alfalfa.  He  cut  three  crops  a 
year.  The  first  crop  yielded  If  tons  per  acre,  the  second  IJ 
tons,  and  the  third  1  ton. 

a.    How  much  was  it  worth  at  $8  a  ton? 

h.    How  many  pounds  of  alfalfa  did  he  obtain? 

15.  At  a  certain  post  office  there  were  sold  in  one  year  12,400 
twenty-five-cent  stamp  books ;  4600  forty-nine-cent  stamp 
books ;  2250  ninety-seven-cent  stamp  books.  How  much  was 
received  for  all  of  them? 

16.  A  merchant  owed  a  debt  amounting  to  $9892.  He 
made  four  payments  of  $1980  each.  How  much  did  he  then 
owe? 

17.  A  river  is  2174  miles  long.  A  steamboat  started  at 
the  mouth  of  the  river  and  traveled  up  stream  for  6  days  at  the 
rate  of  149  miles  a  day.  The  boat  was  then  how  far  from 
the  source  of  the  river?     (Illustrate.) 


28 


REVIEW  AND  PRACTICE 


18.  a,  A  rural  free-delivery  mail  carrier  on  a  certain  route 
is  on  duty  298  days  in  a  year  and  rides  24  miles  each  day. 
How  far  does  he  ride  in  a  year  ? 

5.  He  starts  at  8.15  a.m.  and  returns  home  at  3.15  P.M.  every 
day.     How  many  hours  does  he  spend  on  the  road  in  a  year  ? 

The  number  of  pieces  of  mail  delivered  and  collected  by 
him  in  one  month  was  as  follows: 


Registered  Letters 
Common  Letters  . 
Postal  Cards  .  . 
Newspapers  .  . 
Circulars  .  .  . 
Packages      .     .     . 


Collected 

3 

877 
297 

27 
0 

26 


The  total  number  of   pieces   delivered   was   how   much 

greater  than  the  num- 
ber collected  ? 

d.  If  there  were 
189  families  on  this 
route,  what  was  the 
average  number  of 
pieces  delivered  to 
each  family? 

e.  If  this  carrier's 
salary  is  §900  a  year, 
and  it  costs  him  813 
a  month  to  keep  his 
horse,  how  much  of 

the  salary  is  left  to  pay  him  for  his  year's  work  ? 

/.    The  Post-office  Department  of  the  United  States  pays  the 
carrier's  salary.     He  sells  stamps  to  the  amount  of  $31.50  per 


REVIEW   AND   PRACTICE 


29 


1^^^ 


month,  and  sends  the  money  to  the  Post-office  Department. 
The  amount  received  from  this  mail  route  is  how  much  less  per 
month  than  the  cost  of  delivering  the  mail? 

19.  a.  In  the  year  1905  the  Syracuse  post-office  received 
$411,630.95  for  stamps,  registering  letters,  writing  money 
orders,  and  other  postal  business. 
The  expense  of  carrying  on  the 
post-office  was  f  of  this  amount. 
What  was  the  expense  of  carry- 
ing on  the  post-office  ? 

h.  How  much  did  the  Post-office 
Department  gain  on  account  of 
this  post-office  ? 

c.  In  this  office  722  sacks  and 
pouches  of  mail  were  handled  in 
one  day,  each  sack  and  pouch  con- 
taining an  average  of  154  pieces 
of  mail.     How  many  pieces  of  mail  were  handled  ? 

d.  At  the  same  rate,  how  many  pieces  were  handled  in  a 
year  ? 

20..  a.  Willis  has  15  hens.  They  laid  at  the  rate  of  120 
eggs  apiece  in  one  year.  How  many  eggs  were  laid  by  all  of 
them? 

h.  The  food  for  the  hens  cost  $21.  Willis  sold  the  eggs  at 
an  average  price  of  22  cents  a  dozen.  How  much  more  did  he 
receive  for  the  eggs  than  he  paid  for  the  food  for  the  hens  ? 

c.  What  was  the  profit  from  one  hen  ? 

d.  What  would  be  the  profit  from  75  hens  at  the  same  rate  ? 

e.  Willis  has  a  rectangular  hen  park  30  ft.  long  and  15  ft. 
wide.     How  many  feet  of  wire  netting  will  inclose  it  ? 

21.  From  one  hundred  twenty-two  thousand  take  eighty- 
seven  thousand  ninety-four. 


30  INDICATED  WORK 

INDICATED  WORK 

42.  In  problems  requiring  several  operations,  or  steps,  it  is 
well  first  to  indicate  the  operations  by  means  of  signs;  e.g. 
5208  -T-  3  X  8  means  that  we  are  to  divide  5208  by  3  and  multi- 
ply the  quotient  by  8. 

The  parenthesis  (  )  is  sometimes  used  to  inclose  certain 
numbers  or  expressions  which  are  to  be  taken  together  as  one 
thing ;  e.g.  18  x  (15  + 10)  means  that  18  is  to  be  multiplied  by 
the  sum  of  15  and  10. 

Operations  indicated  within  a  parenthesis  should  always  be 
performed  first;  e.g.  5208  -r-  (3  x  8)  means  that  we  are  to  multi- 
tiply  3  by  8  and  divide  5208  by  the  product. 

When  the  parenthesis  is  not  used,  indicated  multiplication 
and  division  should  be  performed  before  indicated  addition  and 
subtraction;  e.g.  125  +  3x18  —  46^23  means  that  we  must 
multiply  3  by  18,  then  divide  46  by  23,  then  add  and  subtract 
results  as  indicated;  thus,  125  +  3  x  18  —  46  ^  23  =  125  +  54  — 
2  =  177. 

43.  Perform  the  operations  indicated: 

1.  5208-f-3x8 

2.  5208  H-  (3  X  8) 

3.  203x607015-596034 

4.  203  X  (607015  -  596034) 

5.  487 +  598 +  645- (2030 -1435) 

6.  9346  -  (6342  +  347  +  89)  +2349 

7.  9346-6342  +  347  +  89  +  2349 

8.  41983  -  87  X  103  +  47 

9.  (41983  -  87)  X  (103  +  47) 

10.  2310  ^  10  X  7  +  604  X  35 

11.  2310 -i- (10  X  7)  +  604  X  35 


INDICATED  WORK  81 

12.  3056  +  9821-7x48-29 

13.  3056  +  (9821  --  7)  x  (48  -  29) 

14.  (|1.25x6  +  25x|.06  +  2x$.30)-5-24.  See  example  3, 
page  26. 

15.  $738.58 -(1132.75 +  1175.50 +  1216).  See  example  4, 
page  26. 

16.  1216  -^  (35  +  41).     See  example  9,  page  26. 

44.  Indicate  and  find  results: 

1.  1.75  less  the  sum  of  $.32  and  1.18. 

2.  1500  less  $275,  and  th.e  result  added  to  1 132. 

3.  $18  more  than  the  difference  between  $27  and  $425. 

4.  The  product  of  1125  and  8,  divided  by  125. 

5.  The  sum  of  18  yards  and  41  yards  taken  away  from  4 
times  69  yards. 

6.  The  sum  of  498  and  747  divided  by  the  difference  be- 
tween 2342  and  2425. 

7.  George  earns  55  cents  a  day  and  Harry  79  cents.  How 
much  do  they  both  earn  in  the  month  of  October,  allowing  for 
4  Sundays  ? 

8.  The  product  of  162  and  39  divided  by  the  difference  of 
87  and  61. 

9.  Frank's  earnings  for  the  6  days  of  the  week  were  $.43, 

$.59,  $.62,  $.79,  $.38,  $.48.     How  much  more  must  he  earn 
before  he  can  buy  a  $5  set  of  books  ? 

10.  The  quotient  of  12,848  -^  16  is  how  much  less  than  the 
number  of  hours  in  1  year  ? 

11.  A  lady  bought  10  yd.  of  silk  at  $1.10  a  yard  and  2  yd. 
of  silesia  at  25^  a  yard.  How  much  change  should  she  receive 
from  a  20-dollar  bill  ? 


32     SPECIAL  CASES  UST  MULTIPLICATION"  AND  DIVISION 

SPECIAL   CASES   IN    MULTIPLICATION   AND   DIVISION 

PRINCIPLES 

45.  1.  Each  removal  of  a  figure  one  place  to  the  left  increases 
its  value  tenfold;  e.g.  5  =  5;  50  =  5  x  10;   500  =  50  x  10. 

2.  Each  removal  of  a  figure  one  place  to  the  right  diminishes 
its  value  tenfold',  e.g.  500-^10  =  50;  50 -j- 10  =  5. 

46.  Oral 

1.  What  is  the  shortest  way  to  multiply  by  10  ?  By  100  ? 
By  1000?  By  10000?  By  1  with  any  number  of  ciphers 
annexed  ? 

2.8x10  =  ?      8x100  =  ?     8x1000=?     8x10000  =  ? 

3.  Cutting  off  a  cipher  from  the  right  of  a  number  is  the 
same  as  moving  all  the  figures  one  place  to  the  right.  How 
does  it  affect  the  value  of  the  number  ? 

4.  How  many  ciphers  must  be  cut  from  the  right  of  a 
number  to  divide  the  number  exactly  by  100  ?  By  1000  ? 
By  10000? 

5.  Divide  each  of  these  numbers  by  10 : 

50;  600;  5290;  36000;  4500;  321560. 

6.  Divide  each  of  these  numbers  by  100  : 

300;     4500;     6000;     78000;     70000;     831000. 

7.  Divide  9600000  by  10000. 

8.  Multiply  each  of  these  numbers  by  10,  100,  and  1000 : 

7;     61;     20;     310;     402;     910;     653;     1020. 

47.  Written  287 

1.    Multiply  287  by  3700.  3700 

2009 
86l 
1061900  Product 


SPECIAL   CASES  IN  MULTIPLICATION  AND  DIVISION     3B 


2.    Divide  435600  by  1800. 
242   Quotient 
18j9P)4356p 
36 
75 
72 
36 


1800  =  18  X  100. 

Therefore  we    divide    by   100   and 
then  by  18. 

How  do  we  divide  by  100  ? 


3.    Divide  83,645  by  13,000. 
Q^%%%\  Quotient 

78 
6 

Multiple/  and  test  hy  dividing 

4.  432  by  20,100 

5.  69  by  38,000 

6.  420  by  80,000 

7.  1242  by  3020 

8.  5003  by  960 

Divide  and  test  results: 

14.  257,830  by  590 

15.  4410  by  70 

16.  34,376  by  100 

17.  1,333,800  by  1900 

18.  1,308,580  by  260 

19.  572,400  by  3600 

20.  42,978  by  300 

21.  642,359  by  470 


When  we  divide  by  1000,  there  is  a 
remainder  of  648.  When  we  divide 
by  13,  there  is  a  remainder  of  5  in 
thousands'  place.  5000  +  648  =  5648, 
whole  remainder. 

the  product  hy  the  multiplier: 
9.      208  by  6500 

10.  320  by  420 

11.  86  by  12,000 

12.  409  by  30,800 

13.  6900  X  413 


22.  8,205,900  by  4200 

23.  367,298  by  1600 

24.  368,700  by  3600 

25.  496,789  by  420 

26.  805,060  by  3090 

27.  367,059  by  7800 

28.  8,079,600  by  71,000 

29.  4,380,700  by  3210 


34 


IDEAS  OF  PROPORTION 


48.    Oral 


IDEAS  OF  PROPORTION 


Fig.  a. 


Fig.  B. 


1.  Figure  B  is  how  many  times  as  large  as  figure  A  ? 

2.  If  figure  A  is  1  inch  long,  how  long  is  figure  B  ? 

3.  If  A  and  B  are  pieces  of  cloth,  and  A  is  worth  f  5,  what 
is  B  worth  ? 

4.  If  A  is  a  piece  of  land  containing  10  acres,  what  is  B  ? 

5.  If  A  is  a  piece  of  land  worth  $12,  what  is  B  worth  ? 

6.  If  B  is  worth  |60,  what  is  A  worth  ? 

7.  If  A  and  B  are  blocks  of  wood  and  A  weighs  9  pounds, 
what  does  B  weigh  ? 

8.  If  B  weighs  39  ounces,  what  does  A  weigh  ? 

9.  If  A  and  B  are  fields,  and  A  can  be  plowed  in  4  days, 
how  long  will  it  take  to  plow  B  ? 

<5d   ■    66  66  66  66 

10.    a,  10  pears  are  how  many  times  2  pears  ? 
h.    2  pears  are  what  part  of  10  pears  ? 

c.  If  2  pears  cost  3  cents,  10  pears  cost cents. 

d.  Two  pears  are  worth cents,  when  10  pears  are  worth 

25  cents. 


IDEAS  OF  PROPORTIOl^ 


36 


11.  a.    One  dollar  is  how  many  times  one  dime  ? 
h.    One  dime  is  what  part  of  one  dollar  ? 

c.  If  one  dollar  will  buy  30  pencils,  one  dime  will  buy 

pencils. 

d.  If  one  dollar  will  pay  for  70  apples,  one  dime  will  pay 
for apples. 

e.  Frank  can  earn  one  dollar  in hours  if  he  can  earn 

one  dime  in  two  hours. 

/.    If  one  silver  dollar  weighs  one  ounce,  one  silver  dime 
weighs ounce. 

12.  If  John  rides  9  miles  in  2  hours,  in  what  time  can  he 
ride  27  miles  at  the  same  rate  ? 

Analysis :   27  miles  are  3  times  9  miles.     Therefore,  if  John  rides  9  miles 
in  two  hours,  he  can  ride  27  miles  in  3  times  two  hours,  or hours. 

Solve  and  analyze  each  of  the  following  problems  : 

13.  If  a  man's  wages  for  12  hours  are  5  dollars,  in  how  many 
hours  will  he  earn  120  ? 


14.  If  20  men  can  do  a  piece  of  work  in  5  days,  how  long 
will  it  take  10  men  to  do  the  same  ? 

15.  When  75^  will  buy  6  pineapples,  how  much  should  be 
paid  for  2  pineapples  ? 


FACTORS  AND  MULTIPLES 


FACTORS  AND  MULTIPLES 


49.  One  of  the  numbers  that  are  multiplied  to  produce  a  num- 
ber is  a  factor  of  that  number  ;  e.g.  2,  3,  and  5  are  factors  of  30 
because  2  x  3  x  5  =  30. 

50.  A  number  that  exactly  contains  another  number  is  a  mul- 
tiple of  that  number ;  e.g.  21  is  a  multiple  of  7.  It  is  also  a 
multiple  of  3. 

51.  A  number  that  is  composed  entirely  of  whole  units  is  an 
integer ;  e.g.  7,  13,  200.  Can  you  name  a  number  that  is  not 
an  integer  ? 

52.  A  factor  that  is  an  integer  is  called  an  integral  factor  ;  e.g. 
8  is  an  integral  factor  of  56. 

53.  A  number  that  is  not  the  product  of  integral  factors  other 
than  itself  and  1  is  a  prime  number;  e.g.  2,  3,  5,  7,  11,  and  13. 

54.  A  number  that  is  the  product  of  integral  factors  other  than 
itself  and  1  is  a  composite  number;  e.g.  16,  24,  35,  1000. 

55.  A  factor  that  is  a  prime  number  is  a  prime  factor;  e.g, 
13  is  a  prime  factor  of  26. 

A  number  that  is  exactly  divisible  by  2  is  an  even  number; 
e.g.  2,  4,  6,  8,  10. 

A  number  that  is  not  exactly  divisible  by  2  is  an  odd  number ; 
e.g.  1,  3,  5,  7,  11. 

Note. —  In  finding  the  factors  of  a  number  it  is  customary  to  consider 
only  integral  factors. 

56.  Oral 

1.  Give  the  factors  of  21 ;  35 ;  49 ;  77  ;  26 ;  39 ;  34 ;  15 ;  91. 

2.  Name  three  factors  of  30. 

3.  Name  as  many  factors  of  24  as  you  can. 


PRIME  FACTORS  37 

4.  Of  what  numbers  are  7,  2,  and  13  the  prime  factors  ? 

5.  Name  four  multiples  of  9. 

6.  132  i^  the  product  of  12  and  what  other  factor? 

7.  Name  all  the  prime  numbers  smaller  than  50. 

8.  84  is  the  product  of  three  factors.     Two  of  them  are  2 
and  6.     What  is  the  other  ? 

9.  Of  what  number  are  2,  3,  5,  and  7  the  prime  factors? 

10.  Give  the  prime  factors  of  15;  25;  21;  33;  77;  30; 
42;  51. 

11.  5,  2,  and  what  other  number  are  the  prime  factors  of  70  ? 

12.  Give  two  factors  of  30  that  are  not  prime. 

13.  What  even  number  is  prime  ? 

57.    Rule  for  finding  whether  a  Number  is  Prime  or  Composite. 

1.  If  the  given  number  is  odd^  divide  it  by  3. 

2.  if  3  gives  a  remainder^  divide  the  given  number  by  5. 

3.  Continue  this  process,  using  each  prime  number  in  order  as 
a  divisor,  until  an  exact  divisor  is  found,  or  until  the  divisor 
equals  or  exceeds  the  quotient.  If  no  exact  divisor  is  found  until 
the  divisor  used  equals  or  exceeds  the  quotient,  the  number  is  prime. 
Otherwise  it  is  composite. 

e.g.  To  find  whether  71  is  prime  or  composite, 

3)71  5)71  7)71  11)71 

23  —  2  rem.  14  —  1  rem.  10 — 1  rem.         6  —  5  rem. 

Since  the  divisor  11,  is  greater  than  the  quotient  6,  and  no 
exact  divisor  has  been  found,  71  must  be  prime. 

Even  numbers  need  not  be  tested ;  for  no  even  number,  ex- 
cept 2,  is  prime.     Why  ? 


38  PRIME  AND   COMPOSITE  NUMBERS 

58.  In  finding  the  factors  of  a  number,  it  is  useful  to  remem- 
ber that 

a.  A  number  is  divisible  by  2  if  the  figure  in  units'  place  is 
even. 

l.  A  number  is  divisible  by  5  if  the  figure  in  units'  place  is 
0  or  5. 

59.  Find  whether  each  of  these  numbers  is  prime  or  composite : 

1.  143  5.  211  9.  121  13.  231  17.  437- 

2.  123  6.  221  10.  97  14.  161  18.  401 

3.  324  7.  119  11.  213  15.  87  19.  593 

4.  163  8.  208  12.  215  16.  78  20.  395 

60.  Written 

1.    Find  the  prime  factors  of  7020. 


2 

7020 

By  what  kind  of  numbers  do  we  divide?    Why? 

Which  divisors  do  we  use  first? 

What  besides  the  divisors  is  a  prime  factor? 

3  .  5  .  13     Prime  factors,  Ans. 

2 

3510 

3 
3 

1755 

585 

3 

195 

5 

65 

2. 

13 
2.3.3 

2. 

I  the  prime  factors  of: 
120                  8.      45 

14.    3381 

20. 

169 

3. 

42 

9.    189 

15.      667 

21. 

561 

4. 

6Q 

10.    665 

16.      310 

22. 

1001 

5. 

110 

11.    429 

17.     399 

23. 

1265 

6. 

105 

12.    425 

18.    1287 

24. 

682 

7. 

462 

13.    414 

19.     253 

25. 

729 

CANCELLATION  39 

CANCELLATION 

61.  Dividing  both  dividend  and  divisor  by  the  same  number 
affects  the  quotient  how  ? 

462  _  ;2  X  ^  X  7  X  ^  _  ^  ^      . 

We  might  express  this  work  as  follows :  dividing  both  divi- 
dend and  divisor  by  2,  then  by  3,  then  by  11 : 

7 

77 

m 

^  =  7  Quotient 

n 
1 

Taking  out  the  same  factor  from  both  dividend  and  divisor  is 
cancellation. 

62.  Solve  hy  cancellation: 

1.  Divide  86  X  27  X  49  X  38  X  50  by  70  x  18  x  15. 

2.  (28x38x48)^(14x19x24x2x2)=? 

3.  (26  X  5  X  54)  -  (13  X  5  X  6)  =  ? 

4.  What  is  the  quotient  of  36  x  48  x  16   divided  by  27  x 
24  X  8  ? 

5.  Divide  5  x  45  x  7  x  20  by  49  x  5  x  4  x  9. 

6.  Divide  5  X  51  X  7  X  9  X  4  by  17  X  20  X  12  X  7  X  2. 

7.  Divide  25  x  2  x  72  x  14  by  6  x  9  x  120. 

8.  How  many  bushels  of  potatoes  at  50  cents  a  bushel  must 
be  given  in  exchange  for  15  pounds  of  tea  at  40  cents  a  pound  ? 


40  REVIEW  AND  PRACTICE 

«o      r\     7  REVIEW  AND   PRACTICE 

63.    Oral 

1.    Name  the  letters  used  in  Roman  notation  and  give  the 


value  of  each. 

In  finding  the  sums  i 

and  difEerei 

ures  first,  thus : 

36  +  46=  ? 

36  +  40  =  76 

76+    6  =  82  .4ns. 

Say  36,  76,  82. 

2.    Mnd  the  sums 

' 

36  +  47 

89  +  27 

81  +  29 

62  +  38 

76  +  39 

48  +  24 

48  +  53 

36  +  17 

3.    Find  the  differences : 

28-19 

41-14 

31-13 

62-28 

43-16 

97-58 

93- 

-27^ 

=  ? 

93  - 

-20z 

=  73 

73- 

-7  = 

66  Ans. 

Say 

93,  73,  66. 

82  +  69 

78  +  36 

38  +  78 

29  +  92 

29  +  33 

26  +  35 

42  +  71 

42  +  99 

31-14 

45-36 

75-37 

109  -  87 

62-19 

203  - 174 

81-45  76-59                58-29              311-82 

4.  Crive  products  at  sight: 

403  X  10  86  X  200 

86  X  100  15  X  40 

22  X  10,000  18  X  300 

14  X  20  12  X  6000 

5.  Find  results: 
27,000  + 13,000  345,000  ^  100 

218  -  38  6250  -f- 10 

550x100  435-^10 


19 

x40 

16 

x500 

200 

xl90 

403 

x8 

8324 

-f-100 

2800 

^400 

1635- 

4-200 

REVIEW  AND  PRACTICE 


41 


6.  Henry  can  row  a  boat  20  rods  in  a  minute, 
and  Eva  can  row  15  rods  in  a  minute.  If  Eva 
is  60  rods  ahead  of  Henry,  in  how  many  minutes 
can  he  overtake  her  ? 

7.  a.  How  many  strokes  of  a  force  pump  are 
required  to  fill  -J  of  a  tank  that  holds  200  gallons 
of  water,  if  a  pint  is  pumped  at  each  stroke  ? 

h.  How  long  would  it  take  at  20  strokes  per 
minute  ? 

64.     Written 


A  Force  Pump 


Find  sums  and  test  your  work.    Can  you  do  it  in  four  min- 

I  T'V  _  J  1   Jl -1    . 


utes  ?     Do  not  copy  addends. 


a,      49      5. 

235 

c. 

8749 

d.    1346.25 

392 

419 

3254 

29.48 

48 

786 

286 

934.29 

6759 

592 

39 

98.65 

24 

839 

458 

813.78 

864 

496 

3476 

92.48 

9837 

318 

239 

9.62 

481 

745 

8375 

46.78 

28 

932 

468 

932.86 

938 

467 

9628 

^8.93 

2.  Subtract  and  test 

; 

a,    4352      h. 

38290 

c,   4001 

d.    603040 

1987 

8199 

102 

13048 

3.  Divide  and  test: 

a.    153825  by  25. 

h.    49386 

by 

78.   c. 

12634  by  500. 

d,  983,700  by  1500.    e.  863,426  by  19,000.    /.  163,801  by  690. 

4.    2,  3,  5,  7,  11,  13,  and  17  are  the  prime  factors  of   what 
number  ? 


42  LEAST  COMMON   MULTIPLE 

5.  What  prime  factor  beside  19  and  11  has  8987? 

6.  Indicate  the  work  and  solve: 

a.  Divide  by  37  the  result  obtained  by  adding  111  to  the 
product  of  148  and  6090. 

h.  A  merchant  bought  345  pounds  of  wool  of  one  man,  3067 
pounds  of  another,  468  pounds  of  another,  and  384  pounds  of 
another ;  and  sold  ^  of  it  at  27  cents  a  pound.  What  did  he 
receive  for  the  part  sold  ? 

7.  Make  and  solve  a  problem  that  might  be  indicated  thus : 

110.00  -  (1.35  4-12.20  + 16.19  +  1.18). 

8.  Solve  by  cancellation, 

(48  X  36  X  55  X  26)  -^  (12  x  22  x  13). 

LEAST  COMMON  MULTIPLE 

65.  Oral 

1.  3  X  4  =  ?     12  is  what  of  3?     Of  4 ? 

2.  2  X  6  =  ?     12  is  what  of  2?     Of  6  ? 

3.  Name  all  the  numbers  of  which  12  is  a  multiple. 

4.  Define  multiple. 

66.  A  number  that  exactly  contains  two  or  more  numbers  is  a 
common  multiple  of  those  riumbers ;  e.g.  12  is  a  common  mul- 
tiple of  2,  3,  4,  and  6.  36  is  also  a  common  multiple  of  2,  3, 
4,  and  6. 

Can  you  name  any  other  common  multiple  of  2,  3,  4, 
and  6? 

67.  The  smallest  number  that  exactly  contains  two  or  more  num- 
bers is  their  least  common  multiple  (L.  C.  M.);  e.g.  18  is  the 
least  common  multiple  of  3,  6,  and  9.  36  is  a  common  multi- 
ple of  3,  6,  and  9.     Why  is  it  not  the  least  common  multiple? 


LEAST  COMMON  MULTIPLE  43 


68. 

Oral 

Find  the  L.  C.  M.  of. 

1. 

2  and  3 

2. 

2,  3,  and  4 

3. 

4  and  6 

4. 

9  and  6 

5. 

10  and  6 

6. 

8  and  6 

7. 

5,  3,  and  2 

8. 

1,  2,  6,  and  4 

9. 

2,  3,  and  9 

10. 

5,  4,  and  2 

11. 

7,  4,  and  2 

12. 

10,  5,  and  4 

13. 

2,  4,  8,  and  12 

14. 

4,  5,  and  12 

15. 

7  and  8 

16. 

16  and  32 

17. 

2,  3,  6,  and  5 

18. 

4,  9,  3,  and  12 

69.  When  the  least  common  multiple  is  a  large  number, 
the  following  direct  method  is  employed  in  finding  it. 

Let  it  be  required  to  find  the  L.  C.  M.  of  12,  15,  and  18. 

12  =  2  X  2  X  3 
15  =  3  X  5 

18  =  2  X  3  X  3 

What  kind  of  factors  have  we  found  ?  A  number,  in  order 
to  contain  12,  must  have  what  prime  factors?  What  prime 
factors  must  it  have  in  order  to  contain  15  ?    18  ? 

A  number  that  contains  12,  15,  and  18  must  have  how  many 
factors  2  ?     How  many  factors  3  ?     How  many  factors  5  ? 

What  is  the  smallest  number  that  has  the  factors  2,  2,  3,  3, 
and  5  ?     What,  then,  is  the  L.  C.  M.  of  12,  15,  and  18  ? 

The  prime  factors  may  be  easily  found  in  this  way: 

2  112     15     18 

^  — fi — T^ Q  ^^  what  kind  of  numbers  do  we  divide  ? 

3      5      3        2  X  3  ><;  2  ;x;  5  X  3  =  180  L.  C.  M. 


44  GREATEST  COMMON  DIVISOR 

70.  Find  the  L.  C.  M.: 

1.  18,  27,  30  8.  15,  60,  140,  210  15.  10,  15,  6,  14 

2.  9,  12,  18  9.  24,  42,  54,  360  16.  48,  20,  21 

3.  16,  48,  60  10.  25,  20,  35,  40  17.  9,  36,  45,  63,  42 

4.  21,  27,  36  11.  14,  21,  35,  45  18.  25,  15,  30,  50 

5.  36,  40,  48  12.  24,  48,  96,  192  19.  13,  19 

6.  18,  24,  36  13.  15,  18,  20,  60  20.  2,  3,  4,  5,  6 

7.  15,  30,  21,  28     14.  16,  24,  40  21.  7,  8,  9,  10 

GREATEST  COMMON  DIVISOR 

71.  A  number  that  will  exactly  divide  two  or  more  numhers  is  a 
common  divisor  of  those  numbers ;  e.g,  .5  is  a  common  divisor  of 
30,  40,  and  60. 

72.  The  largest  number  that  will  exactly  divide  two  or  more 
numbers  is  their  greatest  common  divisor  (G.  C.T),)\  e.g.  10  is 
the  greatest  common  divisor  of  30,  40,  and  60. 

Note.  —  A  common  divisor  is  sometimes  called  a  common  factor^  and  the 
greatest  common  divisor  is  sometimes  called  the  highest  common  factor. 

73.  Numbers  that  have  no  common  divisor  are  prime  to  each 
other ;  e.g.  13  and  15. 

74.  Oral 

1.  Find  the  G-.  0.  D.  of: 

a.  6,  9,  12  e.  8,  24,  40  L  30,  45,  60 

5.  10,  30,  35  /.  14,  28,  42  /.  18,  27,  36 

c.  2,  10,  16  g.  33,  22,  77  h.  12,  24,  36,  48 

d.  12,  30,  18  h.  21,  27,  39  I.  24,  32,  48 

2.  Name  two  numbers  of  which  7  is  a  common  divisor. 

3.  Name  three  numbers  of  which  9  is  a  common  divisor. 

4.  Name  two  numbers  which  are  prime  to  each  other. 


GREATEST  COMMON  DIVISOR 


46 


5.  What  is  the  greatest  number  that  will  exactly  divide  12, 
30,  and  36  ? 

6.  Name  two  numbers  of  which  11  is  the  G.  C.  D. 

7.  Tell  which  of  these  pairs  of  numbers  are  prime  to  each 
other: 

a.  12  and  7        b.  16  and  20        c.  19  and  21        d.  8  and  15 


75.     Written 

1.   Find  the  greatest  common  divisor  of  336,  504,  and  924. 
336  =  ;2x;2x2x2x^x         / 
504  =  ;2x;2x2         x^x3x7 
924  =  ;2  X  ;2  X  ?         x  7  X  11 

2  X  2  x  3  X  7  =  84  G.  0.  D. 

Factoring  the  numbers  and  selecting  the  common  prime  factors,  we  find 
them  to  be  2,  2,  3,  and  7.  Since  all  of  them  are  factors  of  each  of  the  given 
numbers,  their  product,  84,  is  the  greatest  common  divisor  required. 

The  common  prime  factors  may  easily  be  found  in  this  way: 


2 

336 

504 

924 

2 

168 

252 

462 

3 

84 

126 

231 

7 

28 

42 

77 

4 

6 

11 

2  •  2  •  3  •  7  Common  prime  factors. 


Find  the  a.  0.  D.  : 

2.  63,42 

3.  90,105 

4.  112,  168 

5.  132,156 

6.  40,  60,  80 


8.  36,  48,  24        14.  63,  126,  189 


9.  40,  56,  72 

10.  18,  54,  32 

11.  45,  60,  90 

12.  36,  72,  81 


15.  36,  81,  135 

16.  91,  143,  156 

17.  192,  400,  240 

18.  168,  210,  308,  350 


7.   64,  144,  560      13.  44,  121,  33      19.  1980,  945 


46  REVIEW  OF  INTEGERS 

20.  Find  the  greatest  number  that  will  exactly  divide  189, 
378,  and  504. 

21.  Find  all  the  common  prime  factors  of  360,  540,  and  450. 

22.  Find  the  product  of  all  the  common  prime  factors  of  108, 
144  and  360. 

23.  Find  a  number  that  is  prime  to  210. 

24.  Name  three  numbers  of  which  11  is  the  greatest  common 
divisor. 

REVIEW  OF  INTEGERS 
76.   Oral 

1.  (15-4)x(3  +  2)  =  ? 

2.  15-4x3+2=? 

3.  Numerate  137,640,507,239. 

4.  Name  the  periods  in  the  above  number. 

5.  ReadDCXLIV. 

6.  What  is  the  value  of  ^^? 

7.  35  +  48  =  ?  (35  +  40  =  75;  75  +  8  =  83.    Say  35,  75,  83.) 

In  the  same  way  add  : 

a.  6S  and  29;   b.  58  and  15;  e.  49  and  33;  d.  67  and  24. 

8.  The  product  of  two  factors  is  45.  If  one  factor  is  9, 
what  is  the  other?  If  one  factor  is  15,  what  is  the  other?  If 
one  factor  is  6,  what  is  the  other? 

Which  of  the  factors  given  in  your  answer  is  not  an  integral 
factor? 

9.  The  product  of  three  factors  is  108.  Two  of  them  are  4 
and  3.     What  is  the  other? 


REVIEW  OF  INTEGERS  •  47 

10.  Find  tlie  difference  by  subtracting  the  tens  first : 

a.  64-25;  5.81-32;  c.  $.76  -  $  .28;  (^.  11.27 -1.79. 

11.  24  X  20  =  ?  30,700  -- 100  =  ?  1200  -^  200  =  ? 
22.  235  X  1000  =  ?  208  X  10  =  ?  4000  -5-  400  =  ? 

13.  1,500,000-^1500  =  ? 

14.  Edward  earned  $3  one  week  and  $6  the  next.  How 
much  was  left  after  he  had  spent  |  of  it? 

15.  The  change  for  1.24  from  11.00  is  |.06 +  |.70=$.76. 
Say  6,  76. 

Find  the  change  from  11.00  for: 

a.  $.28      c.  $.18      g.  $.69      ^.$.52      2.  $.79       A;.  $.72 
h.  $.10      d.  $.42     /.  $.37      A.  $.39     j.  $.83       L  $.35 

16.  a.  One  day  is  what  part  of  a  week?  If  a  man  pays  $84 
for  a  week's  travelling  expenses,  they  average  how  much  per 
day  ?     h.  At  the  same  rate,  what  would  they  be  for  11  days  ? 

17.  What  part  of  $5  is  $21?  If  $5  will  buy  20  splint 
baskets,  how  many  such  baskets  will  $  2^  buy  ? 

18.  How  many  times  is  $20  contained  in  $400? 

19.  Whatpart  of  $400is$20? 

20.  A  cent  is  what  part  of  a  dime  ?  7  cents  are  what  part  of 
a  dime  ? 

21.  If  a  dime  will  pay  for  20  steel  hooks,  how  many  such 
hooks  will  7  cents  buy  ? 

22.  6  is  what  part  of  12  ? 

23.  When  a  hardware  merchant  makes  a  profit  of  $1.44  on 
12  window  screens,  what  does  he  make  on  6  of  them  ? 


48  REVIEW  OF  INTEGERS 

77.    Written 

Test  and  time  yourself  on  the  first  eight  examples. 


1.  Add : 

2.    Add: 

3.    Subtract:          4.    Subtract 

287 

627 

807,204                   230,007 

965 

438 

99,197                   150,008 

473 

796 

287      . 
69 

342 
109 

5.    2059x78 

218 
246 

781 
627 

6.    786  X  205 -f- 210 

968 
749 

280 
3578 

7.    302  X  (4780  -  3874) 

421 
372 

642 

7986      . 

8.    346793-^5700 

568 

8144 

9.  A  park  in  the  shape  of  a  rectangle,  135  rods  long  and 
48  rods  wide,  contains  how  many  square  rods  of  land  ?  How 
many  acres  ? 

10.  A  certain  street  is  6  rods  wide.  How  long  must  it  be  to 
contain  77  acres  of  land  ? 

11.  A  farmer  has  20  cows  and  feeds  each  of  them  two  quarts 
of  corn  meal  a  day.  How  long  will  100  bushels  of  corn  meal 
last  them  ?     (Solve  by  cancellation.) 

12.  A  man  owed  his  grocer  f  135.  He  paid  |  of  the  debt  in 
labor  and  the  rest  in  cash.  a.  How  much  cash  did  he  pay  ? 
h.  How  many  days  did  he  work,  if  he  received  $2  a  day  ? 

13.  In  the  year  1905,  1,027,421  immigrants  came  to  this 
country ;  317,000  of  them  settled  in  New  York  State,  222,300  in 
Pennsylvania,  20,000  west  of  the  Mississippi  River,  and  the  rest 


FRACTIONS 


49 


in  other  parts  of  the  country,  a.  How  many  settled  in  other 
parts  of  the  country  ?  h.  The  entire  number  was  how  many 
times  the  number  that  settled  west  of  the  Mississippi  ? 


REVIEW  OF  FRACTIONS 

(^Studied  in  the  Primary  Arithmetic) 


M, 


i 


I        I       I       I 


I       I       I       I       I 


N. 


i 


78.    Oral 

1.  This  figure  shows  that  \  —  ^-^.     What  else  does  it  show  ? 

2.  Make  a  figure  to  show  that  f  =  J. 

3.  Make  a  figure  to  show  that  f  =  ^. 

4.  Make  a  figure  to  show  that  |^  =  f  and  ^  =  f  • 


5 

.    What  is  the  value  of  ^ 

'  ^1  H 

^? 

f?  J^: 

>     1? 

¥? 

6 

.    Express  in  lowest  terms: 

I;  f; 

A;  1 

'   is>   ri' 

A 

'   2^' 

7. 

*  +  i  =  ? 

16. 

1  +  1  = 

? 

25. 

f2- 

f=? 

8. 

*  +  i  +  i 

=  ? 

17. 

i  +  i  = 

? 

26. 

A  + 

J  =  ? 

9. 

f  +  j  =  ? 

18. 

|-i  = 

? 

27. 

tV  + 

t=? 

10. 

i-i  =  ? 

19. 

f  +  f  = 

? 

28. 

T>2  + 

i  =  ? 

11. 

i  +  i  =  ? 

20. 

*-J  = 

? 

29. 

i- 

i  =  ? 

12. 

i-i  =  ? 

21. 

J  +  f  = 

? 

30. 

i  + 

1  =  ^ 

13. 

i  +  i  =  ? 

22. 

J  +  i  = 

? 

31. 

i  + 

i=? 

14. 

i--i  =  ? 

23. 

A  +  i  = 

? 

32. 

1- 

tV  =  ? 

15. 

i  +  i  =  ? 

24. 

l%-i  = 

? 

33. 

f- 

i  =  ? 

50  FRACTIONS 

34.  Harold  bought  a  melon.  He  gave  |-  of  it  to  Clarence  and 
^  of  it  to  Howard.     What  part  of  the  melon  did  he  keep  ? 

35.  A  blotter  is  1^  inches  long  and  3|  inches  wide.  What 
is  the  sum  of  the  two  sides  ?  Of  the  two  ends  ?  What  is  the 
perimeter  ?     Draw  the  blotter,  full  size. 


3.   41Jj  4.     25^ 

268|_        _8^ 

7.  84|  8.  152f 
91^        801-1 


79.  Written 

Add: 

1.  12i 
13i 

2. 

58i 

25j 

28| 

S.  18J 

6. 

325| 

5| 

27f 
41i 

Subtract  and  test  your  work: 

9.  43f 

10. 

85V^ 

29J 

37| 

13.  132f 

14. 

203 

24A 

161 

11.  401f     12.  204-j^ 
123J         131 


15.  83      16.  401  If 

HA  j5|_ 


17.  What  must  be  added  to  5J  to  make  16|  ? 

18.  What  must  be  taken  from  18^^^  to  leave  6|? 

19.  Laurence  bought  a  pencil  6lf  inches  long  and  cut  it  into 
3  pieces,  two  of  which  were  3|  and  2^1  inches  long.  How  long 
was  the  third  piece?  Prove  the  correctness  of  j^our  answer  by 
drawing  a  line  6^|  inches  long  and  cutting  off  pieces  3|  and  2^^ 
inches  long. 

20.  Indicate  by  signs  the  work  required  for  example  19. 


FRACTIONS  61 


FRACTIONS 


80.  One  or  more  of  the  equal  parts  of  a  unit  is  a  fraction;  e.g. 

i'  |!  f ;  *A- 

81.  A  fraction  is  always  an  expression  of  division.  For  ex- 
ample, if  1  inch  is  divided  into  8  equal  parts,  each  part  is  -J  of 
an  inch.  If  a  line  7  inches  long  is  divided  into  8  equal  parts, 
one  part  is  -J  of  an  inch  long.  That  is,  1  in.  -v-  8  =  J  in.,  and 
7  inches  -i-  8  =  |-  inch. 

Take  your  rule  and  draw  a  line  1  inch  long.  Divide  it  into 
4  equal  parts.  How  long  is  one  part?  Draw  a  line  3  inches 
long.  Divide  it  into  4  equal  parts.  Measure  one  of  the  parts. 
3  inches  -^  4  =  ? 

Draw  a  line  5  inches  long.  Divide  it  into  8  equal  parts. 
Measure  one  of  the  parts.  5  inches -^-8  =  ?  3h-7  =  ? 
9^11  =  ? 

82.  The  number  above  the  line  in  a  fraction  is  the  numerator. 
It  is  always  a  dividend.  In  the  fractions  ^,  ^,  ^,  |j,  the 
numerators  are  1,  7,  15,  and  23. 

83.  77ie  number  below  the  line  in  a  fraction  is  the  denominator. 
It  is  always  a  divisor.  In  the  fractions  J,  -|,  ^^^  -||,  the  de- 
nominators are  3,  9,  5,  and  12. 

84.  The  numerator  and  denominator  are  the  terms  of  a  frac- 
tion; e.g.  the  terms  of  -^j  are  7  and  11. 

85.  The  value  of  a  fraction  is  the  quotient  obtained  by  dividing 
the  numerator  by  the  denominator. 

REDUCTION   OF   FRACTIONS 

86.  Changing  the  form  of  a  number  without  changing  its  value 
'ds  reduction  ;   e.g.  8  pt.  =  4  qt.;  $7  =  700  ct. ;   7  ft.  =  2^  yd.; 

^^=3;  i|  =  f;|  =  A- 


52  KEDUCTION  OF  FRACTIONS 

REDUCTION  TO  LOWEST  TERMS 

87.  A  fraction  is  in  its  lowest  terms  when  the  numerator  and 
denominator  are  prime  to  each  other  ;  e.g,  |^,  f^^  ||^. 

88.  Oral 

1.  Dividing  both  dividend  and  divisor  by  the  same  number 
affects  the  quotient  how  ? 

2.  1^  compares  how  with  ^  ? 

3.  -^j  compares  how  with  |  ?     What  did  we  do  with  the 
terms  of  ^^  to  obtain  ^  ? 

4.  Show  by  these  circles  that 


i=h 

t  =  |- 

x\  =  h 

^  =  -1- 

A=l- 

iV  =  t- 

fo  =  h 

5.  How  are  these  fractions  reduced  to  lowest  terms  ? 

6.  Reduce  to  lowest  terms  :  |;  |;  |;  |;  |;  -^;  |;  f;  -fji 

h  f^;  A;  t'j;  ^r'  f^;  ^;  ^;  A;  if;  if;  if;  if;  if;  A; 

IXL.    _8_.      8 
12'     12'     1?- 

89.    Written 

Reduce  ||  to  lowest  terms.  ||  =  ||  =  J  4^«.  We  divide 
both  terms  by  2  and  then  by  3. 

If  we  use  the  greatest  common  divisor  (6),  we  shall  need  to  divide  only 
once,  thus  f  f  =  |. 

Note.  —  We  may  often  save  time  by  remembering  that  an  even  number 
will  never  exactly  divide  an  odd  number.     Can  you  tell  why? 


REDUCTION   OF  FRACTIONS  63 


22. 

^1 

23. 

IH 

24. 

3% 

25. 

T% 

26. 

m 

27. 

m 

28. 

m 

90.  Reduce  to  lowest  terms  : 
1-    If  8-    l¥?  15-    ^s¥j 

3.    M  -10.    5^  17.    11^ 

4-    M  11-    ^J  18.    ^ 

5.    If  12.    Ill  19.    ^ 

6-    M  13-    iifs  20.    iH 

'•    A%  "•    SWV  21.    ^% 

29.  Express  in  lowest  terms  230  -f-  345. 

30.  Express  in  lowest  terms  98  divided  by  392. 

31.  Express  in  lowest  terms  437  -4-  2484. 

32.  Express  in  lowest  terms  the  quotient  of  288  divided 
by  504. 

33.  What  are  the  lowest  terms  of  if  J  ? 

REDUCTION  OF  IMPROPER  FRACTIONS  TO  INTEGERS  OR 
MIXED  NUMBERS 

91.  A  fraction  whose  numerator  is  smaller  than  its  denominator 
is  a  proper  fraction;  e.g.  f,  ^^p  \^,  The  value  of  a  proper 
fraction  is  always  less  than  1. 

92.  A  fraction  ivhose  numerator  equals  or  exceeds  its  denomi- 
nator is  an  improper  fraction,  e.g.  |^,  |,  ^|.  The  value  of  an 
improper  fraction  compares  how  with  1  ?  f 

93.  A  numher  that  is  composed  of  an  integer  and  a  fraction  is 
a  mixed  number;  e.g.  5|,  lOi  201-j\. 


54  REDUCTIOlSr  OF  IMPROPER  FRACTIONS 

94.     Oral 

1.  A  boy  has  two  half  dollars.  That  is  the  same  as  how 
many  whole  dollars  ?  Six  half  dollars  equal  how  many  whole 
dollars  ?     How  do  you  find  it  ? 

2.  Eleven  half  dollars  make  how  many  dollars  and  how 
many  halves  over  ?     How  do  you  find  it  ?     Write  it. 

3.  How  many  quarters  make  a  dollar  ? 

4.  How  many  dollars  are  there  in  8  quarters  ?    40  quarters  ? 

5.  Fifteen  quarters  make  how  many  dollars  and  how  many 
quarters  over  ?     Write  it.     What  do  you  do  to  find  it  ? 

6.  I  =  how  many  whole  ones  ?     |  ?     -^  ?     |  ? 

7.  4=?     |  =  ?     Jj^  =  ?     -V^=? 

8.  A  fraction  is  an  expression  of  what  operation? 

9.  How  may  we  find  the  value  of  a  fraction  ? 
10.    Define  the  value  of  a  fraction. 


Mnd  the  values  of: 

11  I          15.    -\Q-          19.  -2g5-          23.    \3.          27.    2L^Sl       31.    1| 

12.  f          16.    I            20.  Aj^         24.    If          28.    ^|          32.     f| 

13.  f          17.    J^          21.  i^          25.    ^9^       •  29.    ^f          33.    ff 

14.  I          18.    ^^          22.  -^          26.    If^        30.    -J|          34.    -L\^ 

95.  Written 

1.  1|1            3.    ^^  5.    i^^            7.    If^il               9.    %4 

2-  W            4-    -Vf  6.    ^            8.    ^               10.    i||il 

11.  -fj*^                 14.    ^f^  17.     S/ji  20.    A|^A 

12.  il4|i               15.    -V^a  18.    i'jl^  21.    i|-fA 

13.  -^                 16.    ^  19.    2J^^  22.    -4111 


REDUCTION  OF  INTEGERS  AND  MIXED  NUMBERS   55 


REDUCTION  OF  INTEGERS  AND  MIXED  NUMBERS  TO 
IMPROPER  FRACTIONS 


96.    Oral 

1.  How   many   fourths   in  1   circle?     In   2   circles?     In  3 
circles  ?     In  4  circles  ? 

2.  How  many  fourths  in  4|  circles  ?     In  2|  circles  ?     In  3| 
circles  ? 

3.  How  many  eighths   in   1   circle  ?     In    3   circles  ?     In   2 
circles  ?     In  4|  circles  ?     In  2|  circles  ? 

4.  How  do  you  reduce  an  integer  or  a  mixed  number  to  a 
fraction  ? 


Reduce  to  improper  fractions 


5.  IJ 

9. 

3| 

13.  8f 

17. 

8J 

6.  41 

10. 

2| 

14.  4| 

18. 

^A 

7.  3f 

IX. 

4f 

15.  5f 

19. 

8t% 

8.  5i 

12. 

21 

16.  64 

20. 

9i 

97.    TTn^fgw 


1.    Reduce  38J  to  a  fraction. 

38  =  38  X  9  ninths  =  342  ninths. 
342  ninths  plus  7  ninths  =  349  ninths. 
The  work  may  be  expressed  thus:     38|  =  ^^  Ans. 

_9 

342 

7 

349 


56  LEAST  COMMON  DENOMINATOR 

Reduce  to  fractions : 


2. 

9A 

9. 

49i^ 

16. 

19t^ 

23. 

35^ 

3. 

^^ 

10. 

25^ 

17. 

29A 

24. 

191^ 

4. 

25f 

11. 

59A 

18. 

149f 

25. 

203  j8,- 

5. 

16^ 

12. 

67t^ 

19. 

1281 

26. 

98ii 

6. 

23^ 

13. 

89M 

20. 

137^ 

27. 

8711 

7. 

40i 

14. 

131| 

21. 

238iJ 

28. 

138j2^ 

8.    3T^  15.    270||  22.   491^  29.    351if 

LEAST   COMMON   DENOMINATOR 

98.  Fractions  whose  denominators  are  alike  have  a  common 
denominator ;  e.g.  60  is  a  common  denominator  of  g^^,  l|,  and  |J. 

99.  Fractions  having  the  smallest  possible  common  denomi- 
nator have  their  least  common  denominator;  e.g.  ^-^,  ^^,  ^. 

100.  Oral 

1.  We  have  found  that  when  we  add  fractions  having  dif- 
ferent denominators,  we  must  first  change  them  to  fractions 
having  the  same  denominator.  What  shall  we  call  that 
denominator  ? 

2.  Since  the  common  denominator  must  contain  all  the 
given  denominators,  it  must  be  what  of  those  denominators  ? 
(A  number  that  exactly  contains  two  or  more  other  numbers 
is  what  ?) 

3.  The  least  common  denominator,  then,  must  be  which 
multiple  of  the  given  denominators  ? 

4.  Reduce  |,  |,  and  J  to  fractions  having  the  least  common 
denominator. 


LEAST  COMMON  DENOMINATOR 


67 


How  many  12ths  in  1  ?     (12  ---  4  =  3.) 
How  many  12ths  in  |  ?     (^^^  =  — . ) 


How  many  12ths  in  1  ?  (12  ^  6  =  2.) 

How  many  12ths  in  f  ?  ('^^^  =  —^ 

^          ^  V6  X  2   12 ; 

How  many  12ths  in  J  ?  (12  -^3  =  4.) 

How  many  12ths  in  |  ?  f^^  =  -^.^ 

^                 ^  V3x4     12; 


Change   the  following   to  fractions   having   the  least   common 
denominator : 

5-  hi 

6-    hi 

12.    f,  I,  f 
13 


6     1 
6'  2 


14. 

J.f'l 

23. 

h  h  h  i 

15. 

f  f '  f . T^ 

24. 

h  I  \h  i 

16. 

J' !.  f  ^ 

25. 

h  tV'  h  J 

17. 

A'  ¥'  i 

26. 

f  T^I-  |.  Jl 

18. 

f .  I'f '  f 

27. 

f  i^  h  i 

19. 

f.*'* 

28. 

h  f .  i'  ■h 

20. 

f  fA 

29. 

h  h  h  h  tV 

21. 

,f '  h  H 

30. 

h  h  h  1^'  tV 

22. 

i'l^' A 

31. 

i'  ¥'    16'  32'  2 

58  REVIEW  AND  PRACTICE 

101.    Written 

Change  •:j^,  j^^,  Jf  and  ^^  to  fractions  having  the  least  com- 
mon denominator. 


7 

8 

16 

17 

2 

10 

15 

33 

30 

3 

5 

15 

33 

15 

5 

5 

5 

11 

5 

1     1    11      1 

2  X  3  X  5  X  11  =  330  L.  C.  M. 


330 

•^  10  =  33 

7  x33  231 
10  X  33   330 

330 

-f-15=22 

8  x22  176 
15  X  22   330 

330 

-i-  33  =  10 

16x10  160 
33  X  10  330 

330- 

-  30  =  11 

17  X  11  187 

== 

30  X  11      330 

131    116.    110     18X  A7)fi 
330'   330'  330'   330  ^^^' 


Change  to  fractions  having  the  least  common  denominator: 

^-  h  f'  f  6.  1, 1, 1,  J  11.  i|,  ^5_  II 

3.    I'  1%  i  8.    1   |,  I,  ^  13.    1|,  ^5_,  ^ 

^'  fit  9-  I' 14' A  14.  M'l^^'A 

5.    f ,  ^2 ,  if  10.    f ,  ^5^,  f ,  I  15.    ^,  ^^,  J3,  38_ 


102.   Ora? 


REVIEW  AND  PRACTICE 


1.  What  change  should  I  receive  out  of  $2  for  a  purchase 
of  1.50?  8.75?  $.85?  i.45?  $1.25?  11.79?  $.69? 

2.  Henry  bought  a  top  for  3  cents,  some  candy  for  11  cents, 
and  a  pencil  for  7  cents.  What  change  should  he  receive  from 
a  quarter? 

3.  $240  will  buy  how  many  typewriters  at  $60  apiece? 
At  $  80  apiece  ? 

4.  8  cows  at  $  40  a  head  cost  how  much  ? 

5.  What  is  the  cost  of  2  bushels  of  potatoes  at  20^  a  peck  ? 


REVIEW  AND  PRACTICE  59 

6.  800  +  500  +  700  +  1500  =  ? 

7.  Name  the  prime  factors  of  90. 

8.  Tell  the  value  of  -J/;  1^^;  2i;  3^^;  J_2j)_(i. 

9.  Change  to  improper  fractions  8|  ;  2|;  17| ;  5^J;  241; 

10.  What  is  a  fraction  ?  If  you  change  |  to  eighths,  how 
will  its  value  be  affected  ?  How  will  the  number  of  parts  be 
changed  ?     How  will  the  size  of  the  parts  be  changed  ? 

11.  How  does  ^  compare  with  ^  ?     Show  this  by  a  drawing. 

12.  Which  is  larger,  1  of  an  apple  or  ^  of  an  apple  ?  -^-  or  J? 
fori? 

13.  Which  is  greater,  |  or  |  ? 

14.  29  pounds  are  how  many  times  5  pounds?  Compare 
$250  with  $50;  1  qt.  with  1  pt.;  80^  with  20)?^. 

15.  Compare  2  cents  with  50  cents ;  2  gal.  with  3  gal. ;  8  lb. 
with  64  lb. ;  $.25  with  $1.50. 

16.  Find  the  cost  of  24  souvenir  cards  at  the  rate  of  3  for 
5  cents. 

17.  A  windmill  turned  20  times  a 
minute  with  a  certain  wind.  The 
owner  oiled  the  bearings  of  the  mill 
and  then  it  turned  24  times  a  minute 
with  the  same  wind. 

a.  How  many  turns  per  hour  were 
gained  by  oiling  the  bearings  ? 

h.  How  many  times  as  much  work 
did  the  mill  do  after  oiling  as  before 
oiling  ? 

c.  What  part  as  much  work  did  the 
mill  perform  before  oiling  as  after  oiling  ? 


60 


REVIEW  AND  PRACTICE 


103.    Written 


Land  Surfaces  in  Square  Miles 


New  York  .     .     . 

47620 

Rhode  Island  .     .     , 

1080 

Texas     .... 

262290 

Pennsylvania  .     . 

.      44980 

Nebraska    .     .     .     . 

76840 

Connecticut     .     . 

4850 

Delaware    .     .     . 

1960 

Illinois  .... 

56000 

California   .     .     . 

155980 

Montana     .     .     . 

145310 

Kentucky   .     .     . 

40000 

Massachusetts 

8040 

New  Jersey      .     .     . 

7450 

New  Hampshire    . 

9000 

District  of  Columbia 

.     .       60 

Alaska  .     .     (nearly)  570390 

Note  1.  —  Water  surfaces  are  not  included  in  the  above  figures. 

Note  2.  —  While  answering  questions  1-5,  keep  your  geography  before 
you,  open  at  the  map  of  the  United  States.  By  referring  to  the  map, 
estimate  each  answer  before  computing  it,  and  then  compare  your  estimate 
with  the  result  obtained  by  computation. 

1.  a.  Texas  contains  how  many  times  as  much  land  as  New 
York  ?  h.  It  contains  how  many  more  square  miles  of  land 
than  New  York  ? 

2.  Alaska  would  make  how  many  states  the  size  of  New 
Hampshire  ? 

3.  Compare,  by  division,  the  land  areas  of: 

a.  Alaska  and  Illinois. 

b.  Illinois  and  New  Hampshire. 

c.  New  Jersey  and  Pennsylvania. 

d.  Rhode  Island  and  Texas. 

e.  Massachusetts  and  New  York. 
/.  Connecticut  and  California. 

ff,  Montana  and  Delaware. 

h.  Rhode  Island  and  District  of  Columbia. 

4.  Compare,  by  subtraction,  the  land  areas  of: 

a.  Nebraska  and  Pennsylvania. 


ADDITION  OF  FRACTIONS  AND  MIXED  NUMBERS       61 

h.  Delaware  and  New  Hampshire. 

c.  Kentucky  and  Rhode  Island. 

d.  Illinois  and  Massachusetts. 

e.  Texas  and  Alaska. 

5.  a.  Find  which  of  the  columns  of  land  surfaces  (top  of 
page  60)  indicates  the  greater  number  of  square  miles. 

b.  What  is  the  difference  ? 

Make  other  problems  from  the  above  table. 

6.  Find  the  prime  factors  of  1232. 

7.  Reduce  365J  and  66|  to  improper  fractions. 

8.  Reduce  -^^^  to  lowest  terms. 

9.  How  many  15ths  are  there  in  39  ? 

10.  Find  the  value  of  ^OgO-;  %2_4;  J^_t 

11.  How  many  40ths  are  there  in  7|  ? 

12.  7  is  equal  to  what  fraction  having  7  for  a  denominator  ? 

13.  Reduce  to  lowest  terms: 

«•    }lf     *•    iff     0.    Ill     d.  ii,    e.    Uf    /•    ^-h    9-    HI 

14.  Reduce  to  fractions  having  the  least  common  denominator : 

"•      9'    IB'    10       ^-      1?'   21'   35         ^'      11'   3'   8 

ADDITION  OF  FRACTIONS  AND  MIXED  NUMBERS 

104.  A  number  is  in  its  simplest  form  when  it  is  in  the  form  of 
an  integer,  or  a  proper  fraction  in  its  lowest  terms,  or  a  mixed 
number  whose  fractional  part  is  in  its  lowest  terms  ;  e  .g.  18,  |  and 
5^  are  in  their  simplest  forms  ;  ■^,  |^,  -*^  and  8|  are  not  in  their 
simplest  forms.     Why  ? 

Answers  should  always  be  expressed  in  simplest  form,  unless  the 
question  requires  a  different  form. 


105.    Oral 

Add: 

1-    hi 

5. 

ii 

2-    hi 

6. 

hi 

3-    h\ 

7. 

hi 

4-     f.l 

8. 

hi 

62      ADDITION  OF  FRACTIONS  AND  MIXED  NUMBERS 


9.    f,J,3  13.    1|,2^,J 

10.  i,|,  11  14.    3i,lf,5 

11.  21  41  1  15.    f,  8J,  4 

12.  f  j.jj       16.  ^,^,^ 

17.  A  man  paid  I J  for  a  book,  f  |  for  an  inkstand,  and  f  J  for 
writing  paper.     How  much  did  he  spend  ? 

18.  Mary  had  8f ;  her  mother  gave  her  $8 J.  How  much 
had  she  then? 

19.  The  addends  are  7f ,  16 J,  IQlf.     What  is  the  sum  ? 

20.  Mary  walked  5|  miles  on  Monday,  4  miles  on  Tuesday, 
5|  miles  on  Wednesday,  and  as  far  during  the  next  three  days 
as  during  these  days.     How  far  did  she  walk  in  all  ? 

106.     Written 


9      8      6 


3      4      1  f  If  =  2^  S^^- 

2x2x3x3x4  =  144,  L.  C.  D. 

Add  lOf ,  7f ,  and  6|. 

o  ■"       ?  0^  We  add  the  whole  numbers  and  frac- 

•  t  =     •  ^1^  tions  separately,  and  then  unite  the 

6f  = 


sums. 


2^^  =  2^^Sum, 


17. 

6f ,  8|,  f ,  i 

18. 

3,  f ,  h  1 

19. 

f .  4,  f,  li 

20. 

1'  h  iV  t\ 

21. 

6f ,  81  5|,  7| 

22. 

9i,5f9,J| 

23. 

i  f .  %  i 

24. 

6i,  7f ,  91,  45 

SUBTRACTION  OF  FRACTIONS  AND  MIXED  NUMBERS     63 
Add: 

4.  f,  i,  f  12.    f,  1^,  I,  i 

5.  JL,-|,|  13.    J,2|,|,6 
6-    h  i  I'  ^  "•    i'  I'  2i,  tV 

^'     f'  A'  2  0'  2  ^^*     4 9'  y   3'  2T 

25.  What  is  the  sum  of  14f  9|,  IQi  and  12lf  ? 

26.  A  man  travels  25|  miles  on  Monday,  37|  miles  on  Tues- 
day, and  on  Wednesday  as  many  miles  as  on  Monday  and  Tues- 
day.    How  many  miles  does  he  travel  in  three  days  ? 

27.  A  farmer  has  27  J  bushels  of  potatoes  in  one  bin,  133| 
bushels  in  another,  47^^^  bushels  in-  another.  How  many 
bushels  has  he  ? 

28.  How  many  yards  of  cloth  will  I  have,  if  I  buy  123|  yards, 
76|  yards,  and  58|  yards  ? 

29.  2J  yards  of  cloth  are  required  for  a  coat,  IJ  yards  for 
trousers,  and  |  of  a  yard  for  a  vest.  How  many  yards  are 
required  for  the  whole  suit  ? 

SUBTRACTION  OF  FRACTIONS  AND  MIXED  NUMBERS 
107. 


7. 

Oral 

1. 

f-i 

6. 

f-f 

11. 

12 -H 

2. 

«-J 

7. 

T^.-| 

12. 

22 -l| 

3. 

l-i 

8. 

il-f 

13. 

4-lJ 

4. 

i-i 

9. 

7-J 

14. 

^-^ 

5. 

T\-f 

10. 

8-1 

15. 

11 -8f 

64     SUBTRACTION   OF  FRACTIONS  AND  MIXED  NUMBERS 


16.   19|-9f 

20. 

6-i 

24.    121-51 

"•  i-i 

21. 

3i-i 

25.    81 -4J 

18.    i-i 

22. 

15-21 

26.    il-l 

19-    i-i 

23. 

10-f 

27.    1-1 

108.    Written 

From  \^  take  |. 

1  =  1^ 

How  is 

45  obtained? 

■Jl  Difference 

From  29J  take  IS-,;^. 


How  do  we  obtain  |4? 


15J|^  =  15^  Difference 


1.  Take  1  from  f. 

2.  From  Jf  take  ^p 

3.  Find  the  difference  between  l|^  and  ^. 

4.  Take  91|-  from  1781 


5. 

H-i 

13. 

U-i 

21. 

13,^-3^ 

6. 

H-H 

14. 

81-51 

22. 

481^^-232,^ 

7. 

42f-33f- 

15. 

2101 -109f 

23. 

862|-46f 

8. 

198f-49f 

16. 

12i-5i 

24. 

2301-140^ 

9. 

H-H 

17. 

n-H 

25. 

891-431 

10. 

16f-8f 

18. 

461-271 

26. 

807-jV-298} 

11. 

l-i 

19. 

1867f-976| 

27. 

190|  -  28f 

12. 

l-A 

20. 

321  _  26f 

28. 

281^-371 

ADDITION  AND   SUBTRACTION  OF  FRACTIONS  65 

29.  A  piece  of  silk  contains  18J  yd.     How  many  yards  will 
be  left  after  13 J  yd.  are  used  ? 

30.  Mrs.  Brown  bought  4|  yd.  of  broadcloth  and  used  all  but 
1|  yd.     How  much  did  she  use  ? 

31.  Find  the  difference  between  256J  and  149J. 

32.  A  man  bought  a  lot  at  auction  for  $92^  and  sold  it  the 
next  day  for  |105|.     What  did  he  gain  ? 

ADDITION   AND   SUBTRACTION   OF  FRACTIONS 
109.    Written 

1.  From  a  piece  of  cloth  containing  47|^  yd.,  22|  yd.  were 
sold  to  one  lady  and  5|  yd.  to  another.  How  many  yards 
remained  unsold  ? 

2.  A  farmer  sold  a  load  of  hay  for  f  13^"^  and  another  for 
$16|.     He  was  paid  $25.     How  much  was  still  due  ? 

3.  A  lady  paid  $1-^-^  for  a  pair  of  gloves,  $S^  for  an  umbrella, 
and  1292'^Q-  for  dress  materials.  How  much  should  she  have  left 
from  four  ten-dollar  bills  ? 

4.  What  must  be  added  to  the  sum  of  |-  and  10|  to 
make  20? 

5.  From  the  sum  of  109|  and  87|  take  their  difference. 

6.  A  grocer  drew  at  one  time  9|-  gallons  and  at  another 
time  15f  gallons  from  a  tank  containing  44^g  gallons  of  oil. 
How  many  gallons  were  left  ? 

7.  Mary  and  Alice  live  on  Bryant  Avenue,  and  their  school 
is  on  the  same  street,  between  their  homes.  Mary  walks  40^2_. 
rods  to  school,  and  Alice  25J|  rods.  a.  How  much  farther 
does  Mary  walk  than  Alice  ?  b.  How  far  apart  are  their 
homes? 


66  ADDITIOI^  AND  SUBTRACTION  OF  FRACTIONS 

8.  It  took  Mr.  Farmer  S^^  hours  to  plow  a  field,  and  13| 
hours  to  plant  it.  a.  How  much  more  time  was  required  for 
planting  than  for  plowing  ?  b.  How  much  time  was  required 
for  both  ? 

9.  132\-2^3_  +  7|._4i|^  ?  Can  ^ou  find  the  result  in 
two  ways  ? 

10.  a.  i|  +  (li-J)  =  ?     5.    8i|-(li  +  J)  =  ? 

11.  Roscoe  gave  ^  of  his  new  writing  pad  to  his  sister  and 
^  to  his  brother.     What  part  did  he  keep  ?    • 

12.  From  16f  +  12|  +  51  take  18f  +  6|.  .; 

13.  Add  |,  |,  ^^2'  '^^^  A  '  ^^^®^  subtract  IJ  from  the  sum. 

14.  After  reading  g^,  f,  and  1  of  a  book,  what  part  have  you 
yet  to  read  ? 

15.  A  owns  79^5g  acres  of  land,  B  9^^  acres  less  than  A, 
and  C  25^  J  acres  less  than  B.  a.  How  many  acres  has  B  ? 
b.  How  many  acres  has  C  ?  e.  How  many  acres  have  all  three 
together  ? 

16.  From  4223^0-  take  826|. 

17.  Take  8^2  ^^^m  17^^. 

18.  A  has  I641-.  B  has  $37|  less  than  A.  How  much 
money  have  both  ? 

19.  19^g  yards  of  twine  were  cut  from  a  ball  containing 
69^  yards.  The  piece  that  was  left  was  how  much  longer 
than  the  piece  cut  off  ? 

20.  Add  |-,  |,  and  ^J,  and  subtract  the  sum  from  5. 

21.  Find  the  sum  of  |,  ^,  -f^^  and  ■^\. 

22.  Find  the  difference  between  |  and  -^^y. 

23.  A  boy  walked  to  his  grandfather's  in  three  hours,  walk- 
ing -^^  of  the  distance  the  first  hour  and  ^  the  second  hour. 
What  part  of  the  distance  did  he  walk  the  third  hour  ? 


MULTIPLICATION  AND  DIVISION  COMBINED  67 

QUICK  TEST 
110.   1.    39  cents  is  how  much  less  than  one  dollar  ? 

2.  25x20  =  1000-^? 

3.  Which  is  greater,  |-  or  J  ? 

"^       4.  3400  is  the  product  of  100  and  what  other  number  ? 

5.  Express  ||-  in  simplest  form. 

6.  Express  J  as  28ths. 

7.  How  far  will  a  motor  car  run  in  12  hours  if  it  runs 
at  the  rate  of  50  miles  in  4  hours  ? 

8.  What  is  the  L.  C.  M.  of  2,  3,  4,  and  5  ? 

9.  What  is  the  G.  C.  D.  of  21,  35,  and  49  ? 

10.    16  is  the  sum  of  10|  and  what  other  number? 


111. 


MULTIPLICATION  AND  DIVISION  COMBINED 

Written 

/ 

(20^ 
20-1- 

4)  X  (21  -  7)  =  ? 

4  =  5     21-^7  =  3     5x 

3  =  15 

Ans. 

or. 

fxf 

20  x21 
4x7 

=  15     Ans. 

20  and  21  are  dividends  and  4  and  7  are  divisors.  The  result  is  the  same 
whether  we  make  each  division  separately  and  then  multiply  the  quotients, 
or  divide  the  product  of  the  dividends  by  the  product  of  the  divisoi-s.  In 
many  cases  the  latter  way  is  easier,  because  we  may  use  cancellation  ;  e.g. 

5        8 
I       a.(20  +  4)xC21-i-7)  =  (|x|)=?|^  =  15^«s.; 

^7      42 
b.  (18  +  7)  X  (28  +  24)  X  (210  ^  l.S)  =  ^^-^|i2lM=  42  Ans. 

0  ^ 


68  MULTIPLICATION  OF  FRACTIONS 

Find  results  : 

1.  (12-5-ll)x(22--5)x(35-*-6)x(15-f-2) 

2.  (20 -5- 6)  X  (55 -^  10)  X  (42 -11) 

3.  (39 -^  13)  X  (35 -5- 21)  X  (12^  7)  X  (21 -J- 3) 

,     42     36      63 
*•   T^T^li 

5.  (27  -^  18)  X  (35  -5-  75)  x  (25  -^  12)  x  (12  -  7) 

6.  (68 -I- 7)  X  (14  -  8)  X  (35  ^17) 

7.  (52  -^  10)  X  (34  H- 13)  X  (125  -h  10) 

8.  (26 -^  20)  X  (68 -f- 13)  X  (125-^35) 

9.  (70 -f- 17)  X  (68 -5- 24)  X  (35 -i- 7) 

7510898  49241720 

42  ^  26  ^  15  ^^*    56  ^  34  ^  5  ^  3 

12.  Multiply  the  quotient  of  29  divided  by  12  by  the  quo- 
tient of  84  divided  by  29. 

MULTIPLICATION  OF  FRACTIONS 

112.  Any  integer  may  be  expressed  as  a  fraction  by  writing 
it  as  a  numerator  with  1  for  a  denominator^  e.g. ;  5  is  the 
same  as  -^ ;    19  is  the  same  as  -Ij^ ;   |-  x  7  x  ||-  is  the  same  as 

fx^xM- 

113.  The  word  of,  between  fractions^  means  the  same  as  the  sign 
of  multiplication  ;   e.g.    |of|  =  |x|;  |of4x^  =  fx^X^^^. 

114.  An  indicated  multiplication  of  two  or  more  fractions  is 
called  a  compound  fraction  ;  e.g.  f  X  | ;  ^  X  Jf  X  |f ;  f  of  |. 


MULTIPLICATION  OF  FRACTIONS  69 

115.    Written 

1.    Find  the  product  of  f ,  f ,  and  -^q. 

Each  of  these  fractions  indicates  what  operation  ? 

Since  all  the  numerators  are  dividends  and  all  the  denominators  are  divi- 
sors, we  may  find  the  result  by  dividing  the  product  of  the  numerators  by 
the  product  of  the  denominators,  as  in  Article  111,  using  cancellation : 

3 


8 

15 
56 

■    Ans. 

Find  the  products . 

: 

2.    |X| 

8. 

fof^off 

14. 

fxTxif 

3-    Axf 

9. 

ixi\xl- 

15. 

1  of  1  x  14 

4.    ioiJ^ 

10. 

Axfxi 

16. 

Axifx22 

5.    fofjf 

11. 

AxJ^<ixf 

17. 

2xfof  iV 

^-  i^x^ 

12. 

i|x34xf 

18. 

fT0f34x^ 

7.    .|0f|0f| 

13. 

f  ofl5 

Find  the  value  : 

19.    1  of  40 

22.   f  of  328 

25.   ^J  of  342 

20.   fof42 

23.   f  of  721 

26.    If  of  800 

21.   |ofl6 

24.    1  of  90 

27.   -5^  of  2222 

28.  Find  the  areas  of  rectangles  having  these  dimensions : 
a.    I  in.  by  |  in.  e.    f|-  mi.  by  ff  mi. 

h.    f  yd.  by  f  yd.  /.    ^\  mi.  by  |f  mi. 

c,  -^  rd.  by  |-  rd.  ff.      |-  mi.  by  |^  mi. 

d.  f  ft.  by  I  ft.  A.    \i  mi.  by  |f  mi. 

29.  What  is  the  cost  of  J  of  24  quarts  of  milk  at  5|-  cents  a 
quart  ? 

30.  A  grocer  bought  27  barrels  of  apples  and  sold  ^  of  them 
at  $2^  a  barrel.     How  much  money  did  he  receive  ? 


70 


MULTIPLICATTOISr  OF  FRACTIONS 


MULTIPLICATION  OF  FRACTIONS  ILLUSTRATED 


GRAPHICALLY 


Wi 


fo/i 


i  i 

12       I         12 


bcl. 


io/i 


ion 


iofi 


Jon 


ion 


iofi 


Wi         :*• 


's* 


Fig.  1 


h 


How  many  6ths  ? 


116.    Look  at  Fig.  1  and  answer  : 

1.  How  many  12ths  are  there  in  |-? 

2.  How  many  12ths  are  there  in  |^  of  |-  ? 

3.  How  many  12ths  are  there  in  |? 

4.  How  many  12ths  are  there  in  ^  of  J? 
How  many  12ths  are  there  in  J  of  f? 
How  many  halves  are  there  in  |  of  |  ?     f  o^  f  =  ^  =  2  • 
What  else  is  shown  in  this  figure  ? 

Look  at  Fig.  2  and  answer : 

1.  A,  B,  0,  and  D  together  are  what  part 
of  Fig.  2? 

2.  A  is  what  part  of  |  ? 

3.  A  is  what  part  of  Fig.  2  ? 

4.  A  and  E  together  are  what  part  of 
Fig.  2? 

5.  J.is  what  part  of  ^?     J  of  ^  =  ? 

6.    A-\-  B  +  C=  what  part  of  J?     A  +  B+  (7=  what  part 
of  Fig.  2  ?     I  of  1  =  ? 

Note. — Feet  and  inches  are  sometimes  indicated  by  marks,  thus:  7' 
stands  for  7  feet  j  6"  stands  for  6  inches.     What  does  i"  stand  for  ?    |'  ? 


A 

1        1 
B   1    C  !    D 

1        1 
1        1 

E 

1        1 
1        1 

i         1 
!         1 

Fig.  2 


MULTIPLICATION  OF  FRACTIONS 


71 


From  Fig.  3  answer  the  following  questions 

1.  How  many  parts  like  K  are  there  in 
the  square  inch  ?  i 

2.  What  part  of  a  square  inch  is  ^? 

3.  What  is  the  length  of  JT?  The 
breadth?     The  area? 

4.  The  top  row  of  oblongs  is  what  part 
of  the  square  inch?  ^is  what  part  of  that 
row  ?  -^  of  I  =  ? 

5.  The  left-hand  column  of  oblongs  is 
what  part  of  the  square  inch  ?  ^  is  what 
part  of  the  column  ?     ^  of  ^^  =  ? 

In  Figure  4 : 

1.  M  (the  unshaded  part)  is  how  long  ? 
How  wide  ?     What  is  its  area  ? 

2.  jK"  is  what  part  of  the  square  inch  ? 
M  contains  how  many  parts  like  Kl  M 
is  what  part  of  the  square  inch  ? 

What  does  Fig.  5  show  ? 
Draw  on  the  blackboard  a  square  foot. 
,  Divide  two  opposite  sides  into  fourths 
and  the  other  two  sides  into  thirds.     Con- 
nect the  opposite  division  points.     Show 
that:    a.  ixi  =  yV     ^'    i>^i  =  ^=h 


r 


K 


Fig.  3 


K 

^ 

M 

^fe 

W!^/< 

pJ 

WMy///M 

'■Wl/, 

i- 

m 

¥.:.... 

„2^ 

Fig.  4 


Fig.  5 


Show  as  many  other  facts  as  you  can  by  that  figure. 


To  THE  Teacher.  —  Many  exercises  similar  ixD  the  preceding  may  be 
given  to  interest  children  and  make  the  topic  real  to  them.  We  must  re- 
member, however,  that  these  are  mere  graphic  verifications  of  the  rule  for 
multiplication  of  fractions.  They  neither  prove  nor  derive  the  principle. 
The  authority  for  every  operation  in  fractions  is  found  in  the  principles  of 
division  and  the  relation  of  dividend,  divisor,  and  quotientc 


72  MULTIPLICATION  OF  FRACTIONS 

117.  Oral 

1.  How  much  is  1^  of  I  of  an  inch  ? 

2.  Illustrate  that  \oi  ^  oi  an  apple  is  \  of  an  apple. 

3.  Multiply  I  by  i;  i  by  1;  1  by  1;  1  by  1 

4.  Howmuch  is^of  f?     foff?     ^off?     iof^=? 

5.  A  man  owned  f  of  a  farm  and  sold  ^  of  his  share.     What 
part  of  the  farm  did  he  sell  ? 

6.  James  had  i|,  and  John  |  as  much.     How  much  had 
both? 

7.  If  a  pound  of  tea  costs  If,  what  will  \  pound  cost? 

8.  -J  of  |-  of  a  square  yard  is  what  part  of  a  square  yard  ? 
Show  it  by  a  drawing. 

9.  Frank  gave  Harry  |-  of  his  apple  and  Harry  gave  away  J 
of  his  piece.     What  part  of  the  apple  did  Harry  give  away  ? 

10.  Mr.  Greeley,  having  an  acre  of  ground,  took  J  of  it  for  a 
garden.  He  planted  J  of  the  garden  to  potatoes  and  |-  as  much 
to  corn.     What  part  of  an  acre  of  corn  did  he  have  ? 

118.  Mixed  numbers  may  he  reduced  to  improper  fractions 
and  then  multiplied  ;  tJiua^ 

I|x8jx^x4  = 

2 

6    ;I7     5     ^    50    ,.,   , 
gX^x^xf=g-  =  16t^«». 

Written 

1.  4fx5J  4.  35|x27| 

2.  ll,\x7J  5.  2^xl^ 

3.  177fx3  ^                    6.  Find  3^  of  f  of  11  of  8^ 


MULTIPLICATION  OF   FRACTIONS  73 

7.  4ix7|  9.    -3-3^x25fx| 

8.  5fx8fx^  10.   3ix9fx6| 

11.  Multiply  101  by  I  by  f  by  6f. 

12.  Multiply  :  a.    ISJ  by  14f .     h.   16  by  ■^. 

13.  5|x2fx20         15.    9|x^x2|         17.   ^g  x  4  x  51 

14.  7ix5|xf  16.   6|x|iXi\         18.   -f^xS0x5^ 

19.  If  a  man  earns  f  2|  a  day,  how  much  does  he  earn  in  35 
days? 

20.  Multiply  f  by  I  by  J|  by  |l  by  ^\. 

21.  Find  the  cost  of  16  bushels  of  oats  at  37|^^  a  bushel. 

22.  Mrs.  A  buys  S^  qt.  of  milk  a  day.     What  does  she  pay 
for  it  at  5 J/  a  quart? 

23.  Show  by  a  diagram  that  1  of  ^  =  J. 

24.  How  far  can  Joe  ride  in  3|  hours  if  he  rides  9J  miles  an 
hour  ? 

25.  How  many  square  feet  of  floor  are  there  in  a  room  12^ 
ft.  by  71  ft? 

26.  P^ind  the  cost  of  86  cords  of  wood  at  f  4|  a  cord. 

27.  Find  the  value  of  |  of  a  chest  of  tea  weighing  57 J  lb.  at' 
$f  per  pound. 

28.  «.  I  of  f  of  f  of  f  of  1=  ?     h.  -j9^  of  f  of  IJ^  of  i\=  ? 

29.  a.  T\offfof.3-9^oflfoff  =  ?     h.  ioi^^oiiofil^? 

30.  Mr.  Brown  earns  f  60|  a  month,  and  his  son  |  as  much. 
How  much  does  the  son  earn  ? 

31.  At  1121  a  ton,  how  much  will  9^\  tons  of  hay  cost? 

32.  What  will  be  the  cost  of  48|  yards  of  cloth  at  $f  a  yard? 


74 


REVIEW  AND  PRACTICE 


33.  A  man  gave  124^5^.  acres  of  land  to  his  two  sons,  giving 
I  of  it  to  the  elder  and  |  to  the  younger.  How  many  acres 
did  each  receive  ? 

34.  If  it  requires  21|  days  for  a  man  to  dig  a  ditch,  in  what 
time  can  he  dig  |  of  it  ? 


REVIEW  AND  PRACTICE 
119.    Oral 

1.  Read  305,027,503,060. 

2.  Read  XLVI  ;   CXII ;   CCIV  ;  XCIII. 

3.  If  2  sheep  are  worth  $7,  what  are  8  such  sheep  worth? 

4.  If  12  books,  worth  f  8  apiece,  will  pay  for  a  typewriter, 
how  many  books  at  f  6  apiece  would  pay  for  it  ? 

5.  If  I  make  a  purchase  for  $9.15,  what  change  should  I 
receive  for  a  1 10  bill  ? 

6.  Express  in  simplest  form 


_6_.    lA 
10  '      3 


¥;  If;  6^;  Y; 


4.021 


34' 


¥ 


r^ 


7.  What  is  the  area  of  the  top  of  this 
table  ? 

8.  280^70  =  ?    640-^40  =  ? 

9.  Find  the  product  of  32  and  20. 

10.  ?^|=|    i+^  =  ?    |^?  =  f 

11.  Name  two  numbers  that  are  prime 
to  each  other. 


120.     Written 

1.  Express  in  figures  one  hundred  twenty-five  million,  ten 
thousand,  seven. 

2.  Find  the  G.  C.  D.  of  126,  210,  and  294. 

3.  Find  the  L.  C.  M.  of  720  and  216. 


DIVISION  OF  FRACTIONS  75 

4.  Divide  the  product  of  144,  25,  and  56  by  the  product  of 
48,  120,  and  105,  using  cancellation. 

5.  When  potatoes  are  worth  55/  a  bushel,  how  many  bushels 
must  be  given  in  exchange  for  3  jars  of  butter,  each  containing 
33  lb.  at  25  /  a  pound  ?  Indicate  the  work,  and  solve  by  can- 
cellation. 

6.  When  $150  will  buy  189  bushels  of  wheat,  how  many 
bushels  will  i^50  buy?     (|50  is  what  part  of  $150?) 

7.  Express  in  simplest  form;  a.  |f|  b.  ^-^  c.  |^^  d.  17^|f. 

8.  Change  18  to  ninths. 

9.  Reduce  |J^  to  a  fraction  whose  terms  are  prime  to  each 
other. 

10.  How  many  40ths  are  there  in  7|? 

11.  How  many  99ths  are  there  in  89|^? 

12.  A  certain  block  in  our  city  is  \^  of  a  mile  long  and  J| 
of  a  mile  wide.  What  part  of  a  square  mile  of  land  does  it 
contain  ? 

13.  Find  the  area  of  both  sides  of  a  square  piece  of  cardboard 
whose  edge  is  15|  inches. 

DIVISION  OF  FRACTIONS 

121.    Divide  If  by  f. 

Since  ||  is  a  product  and  |  is  one  of  its  factors,  we  may  state 
the  question  thus: 

§5^5     ^^^35^^x_? 
72     8      '        72     8  X? 
In  order  to  find  the   required  factor  we  must   divide   the 
numerator    35   by   5,   and  the   denominator   72   by   8,   thus: 

35-|-5^7 
72-8     9* 


76  DIVISION  OF  FRACTIONS 

That  is  exactly  what  we  should  do  if  the  question  were: 

7 

72^5     '         7;2     ^      9* 
9 

The  latter  method  is  the  more  convenient,  especially  when  the 
numerator  of  the  divisor  is  not  exactly  contained  in  the  numer- 
ator of  the  dividend  or  the  denominator  of  the  divisor  in  the 
denominator  of  the  dividend. 

Therefore,  to  divide  hy  a  fraction  we  interchange  the  terms  qf 
the  divisor  and  multiply. 

122.     Written 

1.  Divide  4|  by  5|. 

2 
(How  do  we  treat  mixed  numbers?) 

2.  Divide  47  by  ^. 

Solution :  47  -^  GJ  =  ^  -^  -V-  =  ^f  x  ^^  =  f |  =  7^3     Ans. 
(How  do  we  treat  integers?) 


3. 

fi-sV 

9. 

HHi 

IS. 

8^A 

21. 

^Hi 

4. 

1  ^i 

10. 

H^A 

16. 

10-i-f 

22. 

H^H 

S. 

iHf 

11. 

^+5| 

17. 

1^14 

23. 

8J^9f 

6. 

A^l 

12. 

■^■^H 

18. 

11^8 

24. 

n-^u 

7. 

if^f 

13. 

il^Si 

19. 

2i^6J 

25. 

il^^ 

8. 

il^T^ 

14. 

2+i 

20. 

^i^H 

26. 

10|-*-4if 

27.  By  what  must  f|  be  multiplied  to  make  f^-? 

28.  One  factor  of  |^  is  J  J.     What  is  the  other? 


DIVISION  OF  FRACTIONS  77 

29.  How  many  pieces  |  of  an  inch  long  can  be  cut  from  a 
wire  that  is  10^  inches  long? 

30.  When  3|  lb.  of  beef  steak  are  worth  57|  cents,  what  is 
the  value  of  one  poUnd? 

423.  Division  of  fractions  is  sometimes  indicated  hy  writing  the 
dividend  above  and  the  divisor  below  a  line.  Such  an  expression 
is  called  a  complex  fraction;  e.g, 

A,   !_,    M    M    and    i^ 
8|'    16'    25'    7f   ^"^^    If -I 

are  complex  fractions.*    Read  each  fraction. 

A  fraction  whose  terms  are  integers  is  a  simple  fraction ;  e.g, 
W  is  a  simple  fraction. 

7 
1.   Reduce  r-r  to  a  simple  fraction. 


81"  3-1-   3~1^26~26*    ^^*' 


_5 

2.    Reduce  45  to  a  simple  fraction. 


6=A^40  =  Ax-l  =  J-     Ans 
40     17  n^'^jZ)     136*    '^^'' 


74 
3.  *  Reduce  ^^  to  its  simplest  form. 


5 

2 


4.  li 

if 

«■'? 

5.    18| 
6 

-8 

78  DIVISION  OF  FRACTIONS 

In  examples  4—13  change  the  given  complex  fractions  to  simple 
fractions  by  performing  the  indicated  divisions  : 

8.  i^       10.  M     12.  i^ 

9.  i        11.  1!     13.  i^L^ 

14.  If  1^  of  an  acre  of  land  is  worth  f  72^,  what  is  the  value 
of  an  acre  at  the  same  rate  ? 

15.  There  are  5|^  yards  in  a  rod.     How  many  rods  in  70|- 
yards  ? 

16.  At  I  6 J  a  ton,  how  many  tons  of  coal  can  be  bought  for 

$73J? 

EXAMPLES  FOR  PRACTICE 
124.    1.    2|  X  I  -f-  IJ  =  ? 

2.  Multiply  l|  by  i|  and  divide  the  product  by  1^. 

3.  a.    14f  -*-  71  =  ?         5.    71  -I-  14|  =  ? 

4.  Change  to  a  simple  fraction  - — \c)\' 

6.  What  is  one  third  of  one  hundred  seventy-five  and  one 
half? 

7.  The  multiplicand  is  1^^  and  the  product  is  2^j.     Find 
the  multiplier. 

8.  Simplify   |L±|. 


AVERAGES  *  79 

9.    How  many  pounds  of  sugar  at  6^  cents  a  pound  will  pay 
for  12 J  dozen  eggs  at  16  cents  a  dozen  ? 

10.  When  15  yards  of  silk  cost  $  16J,  what  is  the  price  per 
yard  ? 

11.  Divide  75f  by  14f . 

12.  Find  the  value  of  ^-li- 

-^2  16 

13.  In  one  month  Mr.  Finlay  earned  $  46|,  his  wages  being 
$  2^  a  day.     How  many  days  did  he  work  ? 

14.  Divide  f  of  |  of  2|  by  |  of  f  of  7. 

15.  $  75  will  pay  for  how  much  corn  at  f  f  a  bushel  ? 

16.  Divide  the  sum  of  4|  and  5|  by  their  difference. 

17.  If  J  of  a  mile  of  telephone  wire  was  divided  into  14 
equal  pieces,  how  long  was  each  piece  ? 

18.  By  what  must  2^  be  multiplied  to  obtain  2^  ? 

19.  By  what  must  3|  be  divided  to  obtain  l^j  ? 

20.  How  many  aprons  can  be  made  from  10|^  yards  of  cloth, 
if  1 J  yards  are  enough  for  one  apron  ? 

21.  Divide  85f  by  14f . 

AVERAGES 


125.  1.  Jacob  weighed  six  of  his 
chickens,  and  found  their  weights  to 
be  68  oz.,  40  oz.,  63  oz.,  47  oz.,  55  oz., 
and  70  oz.  What  was  the  average 
weight  of  the  chickens  ? 


Solution 

68 

oz. 

40 

oz. 

63 

oz. 

47 

oz. 

55 

oz. 

70 

oz. 

6)343 

oz. 

Total  weight. 

57i 

oz. 

Average  weighty  Arts. 

80  AVERAGES 

2.    Harry's  marks  in  spelling  for  a  month  were  as  follows: 

1st  Week        2d  Week        8d  Week        4th  Week 


Monday 

80 

75 

78 

80 

Tuesday 

86 

90 

84 

88 

Wednesday 

83 

82 

90 

92 

Thursday 

88 

80 

86 

96 

Friday 

92 

76 

90 

100 

a^d.    Find  Harry's  average  for  each  week. 

e-i.  Find  his  average  for  all  the  Mondays,  all  the  Tues- 
days, etc. 

j.  How  much  higher  was  his  fourth  week's  average  than  his 
average  for  all  the  Mondays  ? 

k.    On  which  day  of  the  week  did  he  spell  best  ? 

I.    What  was  Harry's  general  average  for  the  month  ? 

3.  Here  are  the  standings  of  seven  girls  in  three  examina- 
tions : 

Marion  Frances  Dorothy  Helen    Jessie    Hazel     Ruth 

Arithmetic,  70         93         98         92        70        83        95 

Geography,  93         90         76         98         90         84         80 

Language,  95        88        95         79        96        85        76 

a.   Which  girl  has  the  highest  average  ? 

5.  What  is  the  average  of  the  class  in  language  ?  c.  In 
arithmetic  ?     d.    In  geography  ? 

e.  Find  the  difference  between  Marion's  average  and  Hazel's. 
/.    Between  Ruth's  and  Jessie's. 

g.    Find  the  general  average  of  the  class. 

h.  Find  the  difference  between  Helen's  average  and  the  av- 
erage of  the  class. 

4.  Our  outdoor  thermometer  indicated  the  following  tem- 
peratures for  the  mornings  of  last  week :  76°,  82°,  80°,  67°,  60°, 


IDEAS  OF  PROPORTION  81 

70°,  81°.  The  week  before  the  record  was  60°,  63°,  58°,  57°, 
70°,  68°,  72°.  For  which  week  was  the  average  temperature 
higher,  and  how  much  higher  was  it  ? 

IDEAS   OF   PROPORTION 

126.  Oral 

1.  2  is  what  part  of  6  ?  If  6  quarts  of  beans  cost  45  cents, 
what  will  2  quarts  cost  ? 

2.  14  is  how  many  times  2  ?  What  will  14  pears  cost  at  the 
rate  of  2  for  5  cents  ? 

3.  18  is  how  many  times  3  ?  If  a  boy  is  paid  20  cents  for  3 
hours'  work,  what  should  he  receive  for  18  hours'  work  ? 

4.  If  a  boy  works  6  days  to  earn  §4,  how  long  should  he 
worl^  to  earn  flO  ? 

5.  What  should  I  receive  for  5  weeks'  work  when  I  earn  $16 
in  5  days  ? 

6.  One  gallon  is  how  many  times  1  quart?  10  gallons  are 
how  many  times  10  quarts  ?  When  10  quarts  of  milk  cost  60 
cents,  what  should  be  paid  for  10  gallons  of  milk  ? 

7.  If  14  five-pound  jars  of  butter  will  last  a  family  a  certain 
time,  how  many  ten-pound  jars  would  last  the  same  time  ? 

127.  Written 

1.  3  is  what  part  of  4  ?  What  should  a  man  pay  for  three 
acres  of  land  when  4  acres  are  worth  i  189  ? 

2.  How  many  bushels  of  wheat  can  be  raised  on  42  acres  of 
land  when  159  bushels  are  raised  on  7  acres  ? 

3.  How  many  tons  of  hay  can  be  bought  for  §3600  when  17 
tons  cost  1300? 


82  REVIEW  AND  PRACTICE 

4.  How  many  10-gallon  cans  may  be  filled  from  a  tank  of 
oil  that  will  fill  155  two-gallon  cans  ? 

5.  Five  is  how  many  times  three  ?     How  far  can  a  man  walk 
in  5  days  if  he  walks  at  the  rate  of  67  miles  in  3  days  ? 

6.  Find  the  amount  of  cloth  needed  for  36  suits  when  17| 
yd.  will  make  3  suits. 

7.  How  many  8-quart  baskets  of  peaches  would  it  take  to 
equal  in  value  5346  bushel  baskets  of  peaches  ? 

REVIEW  AND  PRACTICE 
128.    Oral 

1.  What  is  the  L.  C.  M.  of  8  and  12  ? 

2.  Name  two  other  common  multiples  of  8  and  12. 

3.  Which  of  these  numbers  are  composite  :  15,  13,  29,  36, 
71,  83,  87,  91,  97,  99  ? 

4.  Name  two  numbers  that  are  prime  to  each  other. 

5.  Name  a  number  that  will  exactly  contain  13. 

6.  What  is  the  smallest  number  that  will  exactly  contain  3, 
4,  and  6  ? 

7.  1  of  J  =  ?     1  and  1  =  ?     1  less  J  =  ?     ^  times  \  =  ? 

8.  From  I  take  |-. 

9.  A  farmer  having  60^eep  sold  \  of  them  at  one  time 
and  \  at  another.     How  many  had  he  left  ? 

10.  A  field  of  3|-  acres  was  planted  to  corn  and  potatoes. 
There  were  1|  acres  of  potatoes.  How  many  acres  of  corn 
were  there  ? 


REVIEW  AND  PRACTICE  83 

11.  A  lady  went  shopping  with  f  10.     After  spending  i3|  in 
one  store  and  $5^  in  another,  how  much  money  had  she  ? 

12.  Change  ^|  to  a  fraction  whose  terms  are  prime  to  each 
other. 

13.  Change  4J  to  an  improper  fraction. 

14.  Find  the  value  of  -2^-. 

15.  My  mother  paid  8  cents   for   one    melon,    7    cents   for 
another,  and  10  cents  for  another.    What  was  the  average  cost? 

16.  Show  by  means  of  a  circle  that  l  of  ^  =  1. 

129.    Written 

1.  Draw  a  clock  face,  using  Roman  numerals.    Let  the  hands 

indicate  a  quarter  past  9. 

2.  Find  the  wages  of  8  men  for  5|  days  at  f  3J  a  day. 

3.  15  sheep  at  |2|  apiece  will  pay  for  how  many  yards  of 
cloth  at  1 1  per  yard? 

4.  A  watch  gains  1 J  seconds  every  day.     How  many  min- 
utes does  it  gain  in  the  months  of  June  and  July? 

5.  There  are  609  pupils  in  a  school,  j-  of  whom  are  girls. 
How  many  boys  are  there  ? 

6.  Divide  ^  by  M. 

7.  A  man  spent  |  of  his  money  and  had  $60  left.     How 
much  had  he  at  first? 

8.  a.    15fxl6J  =  ?     h.    16f-v-33i=?     c.    5f-^li  =  ? 

9.  (2j4-f  +  3)^|  =  ? 

10.    If  6^  bushels  of  rye  cost  $5|,  what  is  the  cost  of  1  bushel? 


84 


REVIEW  AiNTD  PRACTlClii 


11.  Change  to  a  simple  fraction  ^ ^ , 

6| 

12.  |78|  will  buy  how  many  barrels  of  flour  at  f  4|  a  barrel? 

13.  Find  the  value  of  2l. 

14.  The  area  of  a  wall  map  is  976J  square  inches.    Its  length 
is  42|^  inches.     Find  its  width, 

15.  An  alley  between  two  houses  is  |  of  a  rod  wide  and  7| 
rods  long.     How  many  square  rods  of  land  does  it  contain  ? 


16.  a.  The  bunches  of  bananas  hanging  in  this  fruit  stand 
contain  respectively  98,  124,  62,  and  140  bananas.  What  is  the 
average  number  of  bananas  per  bunch  ? 

b.  The  two  smaller  bunches  are  red,  and  sell  at  the  rate  of  2 
for  5  cents.     What  are  they  all  worth  ? 

e.    What  will  the  others  bring  at  15  cents  a  dozen  ? 

d.  If  the  four  bunches  were  bought  at  11.09  per  bunch,  what 
will  be  the  entire  profit  on  the  sales  ? 


REVIEW  AND   PRACTICE  85 

17.  Mr.  Scotese,  the  fruit  dealer,  bought  5  bushels  of  apples 
for  $3.25.  They  average  136  apples  per  bushel.  He  sells  them 
all  at  the  rate  of  4  for  5  cents. 

a.    What  is  the  cost  per  bushel  ? 
h.    What  is  received  for  a  bushel  ? 
c.    What  is  the  profit  on  5  bushels  ? 

18.  a.  He  sold  a  dollar's  worth  of  pears  at  the  rate  of  3  for  5 
cents.     How  many  pears  did  he  sell  ? 

h.  If  he  should  put  the  pears  in  baskets,  12  in  each  basket, 
and  offer  them  to  you  at  14  cents  a  basket,  or  at  the  rate  of  4 
for  5  cents,  which  offer  would  you  take  ? 

19.  a.  He  bought  peaches  at  $1  a  basket  and  sold  them  at 
15  cents  a  quart.  There  were  10  quarts  in  a  basket.  What 
was  his  profit  on  5  baskets  of  peaches  ? 

h.  If  9  of  these  peaches  would  make  a  quart,  and  they  were  sold 
at  2  cents  apiece,  what  would  be  the  profit  on  a  basket  of  peaches  ? 

20.  What  was  his  profit  on  50  baskets  of  Delaware  grapes 
bought  at  12|^  a  basket  and  sold  at  18^  a  basket? 

21.  He  bought  chestnuts  at  $4.00  a  bushel  and  sold  them  at 
20  cents  a  quart.     What  did  he  gain  on  a  quart  ? 

22.  Spanish  shelled  peanuts  cost  him  $9.80  a  sack,  each  sack 
containing  140  pounds.  He  sold  them  at  12  cents  a  pound. 
What  was  the  profit  on  a  sack  ? 

23.  He  bought  50  watermelons  at  23  cents  apiece  and  sold 
them  at  40  cents  each.     What  was  his  profit  ? 

24.  He  paid  f  8  a  hundred  for  cantaloupes  and  sold  them  for 
12  cents  apiece.     How  much  did  he  gain  on  a  hundred? 

25.  John  earned  $  6|  one  week,  1 8f  the  second,  and  $  7^  the 
third.     What  were  his  average  earnings  per  week? 

26.  How  many  pounds  of  sugar,  at  b^  cents  a  pound,  are  equal 
in  value  to  6J  dozen  eggs,  at  15  cents  a  dozen  ? 


86  ALIQUOT  PARTS 

27.  Divide  If  of  -^^  by  3f  times  l^j. 

28.  Divide  f  of  f  of  ^  x  3^  by  f  of  if  of  f 

29.  Solve  in  the  easiest  way : 

a.  If  6  acres  of  land  cost  $  438,  what  will  42  acres  cost  ? 
h.  How  many  loads  of  earth  can  be  bought  for  $80  when 
$400  will  buy  1135  loads? 

30.  -^  is  how  many  times  ^^2 

ALIQUOT    PARTS 

130.  One  of  the  equal  parts  of  a  number  is  an  aliquot  part  of 
that  number;  e.g.  8  oz.  is  an  aliquot  part  of  16  oz.  because  8  oz. 
is  ^  of  16  oz. ;  16 J  cents  is  an  aliquot  part  of  100  cents  because 
16|  cents  =  J  of  100  cents. 

Find  the  number  of  cents  in  If;    I  J;   I  J;    ||;    $1;    ||; 

The  answers  you  have  given  are  all  what  kind  of  parts  of  a 
dollar? 

Prove  the  correctness  of  the  following  table  : 

PARTS   OF   A   DOLLAR 

5    cents  =  1 2V  ^H  cents  =  ||- 

61  cents  =  $ Jg  .     37^  cents  =  $| 

8J  cents  =  lyV  ^^    cents  =  |l 

10    cents  =  $^^  62J  cents  =  if 

12 J^  cents  =  -I  J  66|  cents  =  If 

16f  cents  =  i^  75    cents  =  $f 

25    cents  =  $  J  87^  cents  =  $f 

Which  column  in  the  table  gives  aliquot  parts  ?  This  table  should  be 
committed  to  memory  like  the  multiplication  table,  because  its  use  will 
shorten  many  problems,  e.g.  33  books,  at  $  .16|  each,  will  cost  33  x  $^  =  $  5^. 


i 


ALIQUOT  PARTS  87 

131.  Oral 
Multiply  : 

1.  121-  cents  by  16  ,  7.  37J  cents  by  8 

2.  16f  cents  by  12  8.  50    cents  by  15 

3.  25    cents  by  20  9.  62^  cents  by  8 

4.  33^  cents  by  27  10.  66f  cents  by  9 

5.  6i  cents  by  16  11.    75    cents  by  4 

6.  81  cents  by  24  12.    87J  cents  by  8 

13.  What  is  the  cost  of  : 

16  pounds  of  bacon  at  12|  ^  a  pound  ? 
16  balls  at  50^  each? 
36  yards  of  ribbon  at  331^  a  yard  ? 
36  pounds  of  candy  at  25 j^  a  pound? 
8  pounds  of  tea  at  62|  ^  a  pound  ? 

14.  When  4  geographies  cost  $  3,  what  is  the  cost  of  one  ? 
Of  9?  Of  11?  Of  15?  (There  is  an  easier  way  to  find  the 
cost  of  20  geographies.     What  is  that  way?) 

15.  At  $.75  apiece,  what  must  be  paid  for  3  chairs  ?  4  chairs? 
6  chairs  ?  16  chairs  ?  40  chairs  ? 

132.  Written 
What  is  the  cost  of  : 

166  pounds  of  pork  at  121  cents  a  pound  ? 

348  pounds  of  veal  at  16|  cents  a  pound  ? 

265  boxes  of  strawberries  at  25  cents  a  box  ? 
1215  yards  of  flannel  at  33|-  cents  a  yard  ? 
3580  pounds  of  honey  at  20  cents  a  pound  ? 

748  pounds  of  tea  at  50  cents  a  pound  ? 


88  DECIMALS 

Oral 

1.  At  25^  a  pound,  how  many  pounds  of  butter  can  be 
bought  for  ^8  ?  (How  many  pounds  can  be  bought  forfl? 
For  $8?) 

Divide : 

2.  $3  by  331/  5.  $9  by  121^  8.  $6  by  331/ 

3.  15  by  25/  6.  II  by  6^  9.  |4  by  121/ 

4.  12  by  81/  7.  110  by  50/  lo.  $2  by  25/ 

11.  $3  divided  by  81/ =? 

12.  At  25  cents  apiece,  how  many  hats  can  be  bought  forf  6? 

13.  At  25  cents  a  pound,  how  many  pounds  of  cheese  can  be 
bought  for  15? 

14.  At  16|  cents   a   dozen,  how   many   dozen  eggs   can  be 
bought  for  14? 

15.  How  many  pounds  of  beef  can  be  bought  for  $4  at  16|/ 
a  pound  ? 

16.  At  33 J  /  a  yard,  how  many  yards  of  linen  can  be  bought 
for  1 10? 

17.  How  many  penknives  can  be  bought  for  16  at  33|^  cents 

iece  ? 

18.  24  X  121/  =  ?  19.   f  24  ^  121/  =  ? 


apiece  ? 


DECIMALS 

133.  Each  removal  of  a  figure  one  place  to  the  right  affects 
its  value  how  ? 

We  have  used  this  principle  thus  far  in  dealing  with  in- 
tegers only ;  but  it  holds  true  also  for  numbers  smaller  than 
one.     Thus, 


DECIMALS 

5000. 

.■ 

500.          =  5000. 

J- 10 

50.          =  500.          -^10 

Moving  5  to  the  right . 

5.          =     50. 
.5       =       5, 

hlO 
-10  =  ^ 

\: 

.05     =         .5       - 

-10=ifo 

.005   =         .05     H 

-l^  =  ToV 

.0005=         .005   H 

-10  =  Tol( 

734.       =  7340.      -- 10 

Moving  all  the  figures 

73.4     =    734.      -^10  =  73^^ 

to  the  right 

7.34   =     73.4   -f- 10  =7^3^ 

.734=       7.34-: 

L0  =  3^,V 

89 


Notice  that  we  place  the  decimal  point  (.)  at  the  right  of 
units'  place.  This  shows  where  the  integer  ends  and  the  frac- 
tion begins.  The  places  at  the  right  of  the  decimal  point  are 
called  decimal  places  and  are  named  much  like  those  at  the 
left,  thus  : 


•♦-• 

w 

(A 

4>' 

Vt 

JZ 

T3 

C 
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4638-754966 


.7  is  read  seven  tenths. 
.75  is  read  seventh-five  hundredths. 
.754  is  read  seven  hundred  fifty -four  thousandths. 
.  7549  is  read  seven  thousand  five  hundred  forty-nine  ten-thou- 
sandths. 


90  DECIMALS 

Oral 

1.  Read  .8;  .49;  .786;  .4923;  .56249;  .38T654. 

2.  Read   500;  50;  5;  .5;  .05;  .005;   .0005;  .00005. 

3.  Read  .25;   .36;   .39;   .47;   .365;   .3;   .7;  .403;  .07;  .009. 

4.  How  many  cents  make  one  dollar  ?  What  part  of  a 
dollar  is  one  cent  ?  7  cents  ?  19  cents  ?  41  cents  ?  97  cents  ? 

5.  We  write  25  cents,  $.25,  because  it  is  25  hundredths  of 
a  dollar.  One  dime  is  1  tenth  of  a  dollar.  How  would  you 
write  it  ?  Write  on  the  blackboard  ;  5  dimes  ;  7  dimes  ;  8 
dimes  ;  3  dimes. 

6.  4321.648  is  read  4321  and  648  thousandths. 

Notice  that  and  is  used  between  the  integer  and  the  fraction. 
Read:  6.42;  17.5;  4.23;  583.97;  .640;  7.640;  439.018; 
9341.215;  68.43;  70.9;  893.047;  903.03;  642.008;  95.249. 

Read:  3.4;  .0034;  .987;  2048.017;  .00315;  200.02. 

Write  in  words  on  the  blackboard :  35.08 ;  6.5  ;  .084  ; 
.6082;  235.235;  64.105;  308.02;  56.081;  30.130. 

134.  The  product  of  equal  factors  is  a  power : 

e.g.      4  is  a  power  of  2  because      2x2  =4 

8  is  a  power  of  2  because      2x2x2       =8 
81  is  a  power  of  3  because      3x3x3x3  =  81 
100  is  a  power  of  10  because  10  x  10  =  100 

Name  3  other  powers  of  10. 

135.  A  fraction  whose  denominator  is  10  or  a  power  of  10  is  a 
decimal  fraction  ;  e.g.  -f^,  -^^-q,  .25,  .3421.  Only  decimal 
fractions  can  be  expressed  by  use  of  the  decimal  point  as  in  the 


DECIMAL  FRACTIONS  91 

last  exercise.  When  a  decimal  fraction  is  thus  expressed,  how 
may  we  tell  what  the  denominator  is  ?  Name  some  decimal 
fractions  not  given  here. 

136.  A  number  that  is  composed  of  an  integer  and  a  decimal 
fraction  is  called  a  mixed  decimal ;  e.g.  2.5  ;  130.35  ;  21.007. 

137.  A  fraction  that  is  expressed  hy  writing  the   numerator 
above  ayid  the  denominator  below  a  line  is  a  common  fraction;  e.g. 


h  I'  9'  if'  I'V- 

138.    Change 

to  the  decimal  form: 

1-  i¥(y 

6.    fi^^                 11.   ^^ 

16.    A 

2-  t¥o 

'•  m               12.  ^JjV 

"•    iV 

3-    A% 

8-    h                      13.    ,^ 

18.    5^^ 

*•    A 

9-    lb                    "•   T^% 

19.   8J^ 

5-    'Uo       ' 

10-    1^7                 15-   1% 

20.  64^Vo^ 

Change  to  common  fractions  and  read: 

21.    .36 

26.   .485 

31.  5.6 

22.    .7 

27.    .016 

32.  5.06 

23.    .125 

28.    .16 

33.  5.006 

24.    12.2 

29.    .06 

34.  5.600 

25.   6.25 

30.    .6 

35.  5.060 

Write^  first  as  common  fractions^  or  mixed  numbers,  then  as 
decimals : 

36.  Four  tenths. 

37.  Seventy-five  hundredths. 

38.  One  hundred  twenty-five  thousandths. 


92  .  WRITING  DECIMALS 

39.  Sixteen,  and  forty-eight  hundredths. 

40.  Twelve,  and  four  tenths. 

41.  Six  tenths. 

42.  Six  hundredths. 

43.  Six  thousandths. 

44.  How  many   decimal    figures    are   required    to   express 
thousandths  ?     Hundredths  ?     Tenths  ? 

45.  Read  the  numerators  only  in  examples  36  to  43. 

Write  the  following  as  decimals^  and  read  the  numerator  and 
denominator  of  each : 

46.  Two  hundred  eighty-two  thousandths. 

47.  Fifty-six  hundredths. 

48.  Seven  tenths. 

49.  Six  hundred  thousandths. 

139.    Oral 

1.  What  part  of  10  units  is  1  unit  ? 

2.  What  part  of  1  ten  is  1  unit  ? 

3.  What  part  of  2  hundreds  is  2  tens  ? 

4    In  the  number  555,  what  is  the  value  of  the  first  5  at  the 
right  ?     The  second  5  ?     The  third  5  ? 

5.  Upon  what  does  the  value  of  any  figure  depend  ? 

6.  In  the  number  555,  the  value  of  tho  first  5  is  what  part 
of  the  value  of  the  second  5  ? 

7.  -f^  is  what  part  of  2  units  ? 

8.  In  the  number  5.5,  the  value  of  the  right-hand  5  is  what 
part  of  the  value  of  the  left-hand  5  ? 


READING  AND  WRITING  DECIMALS  98 


9.    In  the  decimal  .555,  what  is  the  value  of  the  first  5  at 
the  right  ?     The  second  5  ?     The  third  5  ? 

10.   Name 
.225;  .3478; 

the  denominator   of    .6  ;   .17 
.06;   .049;   .207;  .3007. 

;   .105;   .006;  .05; 

11.   Read  the  numbers  in  question  10. 

12.   Read: 

a.  .368 
h.  .894 

j.  37.005 
Tc.  25.2036 

s,   .421 
^^.  .691 

c.  .5328 

I.  38.000006 

u.  .81 

d.  .2053 

m.  .498869 

i;.   .637f 

e.  25.623 

n,  4.9836 

«^.  .4378^ 

/.  7.0063 

0,  49.836 

X.  .809Jj- 

g.  28.3005 
h.  .28962 
i.  15.605 

j9.  498.36 
g.  .000400 
r.  .0004 

y.   .430^ 
z.  .6842J 

TTn'^e  deciinally: 

1.  Eight  tenths. 

2.  29  hundredths. 

3.  Sixteen,  and  284  thousandths. 

4.  4584  ten-thousandths. 

5.  Twenty -five  hundredths. 

6.  Twenty-five  thousandths. 

7.  Twenty-five  ten-thousandths. 

8.  Twenty-five  hundred-thousandths. 

9.  Twenty-five  million ths. 
10.  1650,  and  464  thousandths. 


94  WRITING  DECIMALS 

11.  One  thousand  one,  and  36  hundred- thousandths. 

12.  Sixteen,  and  six  thousandths. 

13.  Seven  hundred  eighty-four  millionths. 

14.  Twelve  hundred-thousandths. 

15.  Seventy-five  ten-thousandths. 

16.  Seven  hundred  five  thousandths. 

17.  Seven  hundred,  and  five  thousandths. 

18.  Four  thousand  three  ten-thousandths. 

19.  Four  thousand,  and  three  ten-thousandths. 

20.  Twenty-four,  and  five  hundred-thousandths. 

21.  Seventy-one,  and  seven  hundred-thousandths. 

22.  Four  hundred  thirty-five,  and  four  thousandths. 

23.  Eight  thousand  three  hundred  forty-one  ten-thousandths. 

24.  Ninety-nine,  and  eighty-six  thousandths. 

25.  Seventy-eight,  and  four  thousandths. 

26.  Nine  thousand  seven,  and  two  hundred  seven"  ten-thou- 
sandths. 

27.  One,  and  one  hundred  thousandths. 

28.  One  thousand  one,  and  one  hundred  one  thousandths. 

29.  Ten,  and  ten  ten-thousandths. 

30.  One  hundred,  and  one  hundred  ten  thousandths. 

31.  One  hundred  one,  and  one  hundred  ten-thousandths. 

32.  One  thousand  one  ten-thousandths. 

33.  One  thousand,  and  one  ten-thousandth. 

34.  Two  hundred  seven  thousand,  and  two  hundred  seven 
thousandths. 

35.  Six,  and  six  thousand  ten-thousandths. 


ADDITION  AND  SUBTRACTION  OF  DECIMALS  95 


ADDITION   AND    SUBTRACTION   OF   DECIMALS 

140.  Since  decimal  figures  increase  in  value  from  right  to 
left,  like  the  figures  in  whole  numbers,  we  may  add  and  subtract 
decimals  as  we  add  and  subtract  whole  numbers,  taking  care  to 
write  them  so  that  the  decimal  points  are  all  in  a  column,  thus  : 

4.375  391.42 

.35  165.70316 

28.3065  225.71684  Difference 

351.294 
384.3255  Sum 

The  vacant  places  in  the  addends  and  in  the  minuend  are  treated  as  if 
they  were  occupied  by  ciphers. 

Add: 

i   1.  2.2  2.  3.25  3.  4.5  4.   .004 
34.5          7.163          .168         4.1 
79.89        15.0032        2.12         16.1563 


5.  .175+1.75  +  17.5  +  175. 

6.  145 +  14.5 +  1.45 +  .145 +  .0145. 

7.  3.2 +  14.0063 +  .006  + 25.384 +  .1. 

8.  .8 +  .446 +  59.3 +  2.575 +  1.0056 +  .3. 

9.  1.45  +  2.365  +  96  +  .96  + 15.863  +  4.3  +  .0004. 

10.  446  +  44. 6  +  37562  +  9  +  .8  +  .  321  +  .16. 

11.  21.0005  +  . 3842  +  . 1  +  .005  +  3.6  +  .158. 

12.  1.0006  +  2001.1  +  .003  +  ^.b  +  11.1111. 

13.  205.07  +  301.2  +  687.9124  +  83.045  +  200. 

14.  .308  +  308.  +  8.09  +  9.0786  +  .859. 

15.  2378.  +  23.50  +  .890  +  .089  +  1.0886. 


96  ADDITION  AND   SUBTRACTION  OF  DECIMALS 


.41 

.    1.    Subtract : 

a. 

24.3                 h. 

2.86 

C.    4. 

d. 

2.46 

4.5 

1.325 

1.15 

.005 

2. 

7 -.15 

7. 

29.325-15.14 

3. 

1-.004 

8. 

3.852 -.125 

4. 

13-2.1 

9. 

1.1111 -.0011 

5. 

3.256-1.05 

10. 

500- .05 

6. 

256.1-1.256 

11. 

25.3894-15.005 

-  • 

12.  From  twenty-eight,  and   twenty-five   thousandths  take 
fourteen,  and  twenty-five  hundredths. 

13.  From  one  tenth  take  one  thousandth. 

14.  a.  Which    is    the    greater,   fiJty    thousandths    or    five 
hundredths  ?     h.  Three  tenths  or  three  hundred  thousandths  ? 

15.  Take  one  thousandth  from  one  thousand. 

16.  From  5  hundred  take  5  hundredths. 

142.    Find  results  : 

1.  175  -  30.23. 

2.  .015  +  1.05  +  .57  +  5.7  +  1.04  +  .0045  +  75.36. 

3.  50.4  -  .504. 

4.  25.006  +   200.00008  +  6.00005  -f  49.005  +  300.059. 

5.  2.005  4-  5.5  4-  25.010  -  3.2045. 

6.  Find  the  sum  of  two  hundred  forty,  and  four  hundred 
fifty  thousandths  ;  thirty-four,  and  three  hundredths ;  six  hun- 
dred four,  and  six  hundred  four  ten-thousandths  ;  fifty,  and 
five  tenths. 

7.  A  boy  had  two  balls  of  kite  string.  One  contained 
145.3025  yards  and  the  other  84.3502  yards.  He  made  a  kite 
string  200.02  yards  long.     How  much  string  had  he  left  ? 


DIVIDING  BY  MtlLTIl^LES  Oi"  t:EN  97 

8.    Subtract: 

a.    32.854  c.    86.2  e.    21.101 

9.378  43.948  7. 


b.    19.042  d,    36.015  /.     28.78 

16.854     .  24.008  21.987 

9.  There  are  four  villages  on  the  same  road.  From  the 
first  to  the  second  is  8.46  miles;  from  the  second  to  the  third, 
10.5  miles;  from  the  first  to  the  fourth,  25  miles.  Make  a 
picture  of  the  road  and  find  the  distance  from  the  third  to  the 
fourth  village. 

MULTIPLYING   AND   DIVIDING   BY   MULTIPLES   OF   TEN 
143.     Oral 

1.  How  can  we  multiply  decimals  by  10  ?  By  100  ?  By 
1000  ? 

2.  Multiple/  by  10  : 

5.25;  .06;  3.7;  593.207;  6.800;  9.16;  82;  420;  .035;  .0061. 

3.  Multiply  by  100: 

61.843;   3.215;    75.16;  3.18;   .65;   2.3;  5;   520. 

4.  Multiply  .0612  by  10 ;  by  100  ;  by  1000  ;  by  10,000 ;  by 
100,000 ;  by  1,000,000. 

5.  By  what  must  .0503  be  multiplied  to  obtain  .503?  503.  ? 
50.3?  5030.? 

6.  Moving  the  decimal  point  one  place  to  the  left  is  the 
same  as  moving  all  the  figures  of  the  number  one  place  to  the 
right.  For  example,  moving  the  decimal  point  one  place  to 
the  left  in  the  number  42.3,  it  becomes  4.23.  How  does  this 
affect  the  value  of  the  number  ? 


98  MULTIPLYING  BY  MULTIPLES  OF  TEN" 

7.  What,  then,  is  the  easiest  way  to  divide  a  decimal  by  10  ? 

8.  Divide  by  10  : 

35.;  247.;  385.;  16.;  24.3;  2.59;  347.69;  8.137;  42.69; 
394.68;  .725;   .042. 

9.  How  may  we  divide  decimals  by  100  ?    By  1000  ?     By 
10,000  ? 

10.  Divide  hy  100  : 

3567.8;  937.;  635.25;  42304.;  687.96;  485.03. 

11.  Divide  hy  1000  : 

986;  5321;  63,485;  983.7;  4284.25. 

12.  Divide  hy  10,000  : 
389,076;  42,831;  68,379.5;  425. 

13.  By  what  must  8193  be  divided  to  obtain  819.3  ?  8.193  ? 
81.93?     .8193?     .08193?     .008193? 

14.  How  must  the  decimal  point  be  moved  to  change  tenths 
to  hundredths  ?  Thousandths  to  tenths  ?  Thousandths  to 
millionths  ? 

15:    Divide  as  follows  : 

64.2  by  10;  83.75  by  10;  63.59  by  100;  4251  by  10,000; 
33  by  1000;  5  by  100. 

16.  What  is  the  effect  of  moving  the  decimal  point  three  places 
to  the  left  ?  One  place  ?  Two  places  ?  Four  places  ?  Six  places  ? 

17.  Find  results : 

3.5x10;  6.5-^10;  83-^1000;  .987-^10,000;  .8432  x 
100,000. 

18.  What  is  the  effect  of  moving  the  decimal  point  to  the 
left  one  place  ?  To  the  right  two  places  ?  To  the  left  four 
places  ?     To  the  right  six  places  ? 

19.  How  may  we  multiply  a  decimal  by  10,000  ?  Divide  a 
decimal  by  1000  ?     Multiply  a  decimal  by  100,000? 


MULTIPLICATIO^r  OF  DECIMALS 

MULTIPLICATION   OF   DECIMALS 
144.    Multiply  6.41  by  3.2. 
6.41  =  641  --  100 
3.2    =    32-5-10 
6.41  X  3.2  =  641  X  32  +  100  -*- 10 


641  6.41 

32  3.2 


1282  1282  — -_^ 

1923  1923  \4 


20512  =  641  X  32        20.512  =  641_x  32  -^  100  -$- 10 

How  did  we  divide  by  100  and  by  10  ? 

From  this  we  see  that  to  multiply  decimals  we  multiply  the 
factors  as  whole  numbers  and  point  off  in  the  product  as  many 
decimal  places  as  there  are  in  both  factors  ; 

e,g,   2.8  1.25  .005  25 

8  .6  m  jM 

22.4  .750  .00015  1.60 

Find  the  products  : 

1.  .18  X  .15  8.  13.3  X  1.3 

2.  1.0005  X  .2  9.  100  X  .01 

3.  2.5  X  .06  10:  100.56  x  .0005 

4.  m  X  .005  11.  25.32  X  1.05 

5.  .005  X  1.6  12.  2.84  X  .25 

6.  25.05  X  1.15  13.  3.28  x  1.125 

7.  2.863  X  100  14.  1.111  X  1000 


0 


100  DIVISION  OF  DECIMALS 

145.  Oral 

1.  One  of  the  factors  has  two  decimal  places  and  the  other 
has  five.     How  many  decimal  places  has  the  product  ? 

2.  When  there  are  five  decimal  places  in  one  factor  and  one 
in  the  other,  how  many  are  there  in  the  product  ? 

3.  When  there  are  eight  decimal  places  in  the  product  and 
five  in  one  factor,  how  many  are  there  in  the  other  factor  ? 
When  there  are  six  in  the  product  and  four  in  one  factor? 
Five  in  the  product  and  three  in  one  factor  ? 

DIVISION  OF  DECIMALS 

146.  Divide  10.96516  by  4.67. 

2.348    Quotient 

4.67)10.96-516 

934  We  first  divide  as  in  whole  numbers.     Since  the 

-— — •  dividend  is  a  product  and  the  divisor  one  of  its  fac- 

tors, the  other  factor,  or  quotient,  contains  as  many 
1401  decimal  places  as  the  number  of  decimal  places  in 

2241  ^l*®  dividend,  less  the  number  of  decimal  places  in 

the  divisor. 

Mistakes  may  be  avoided  by  observing  the  fol- 
lowing 


1868 


3736 
3736 


RULE  FOK  PLACING  THE  DECIMAL  POINT 
When  the  divisor  is  an  integer^  place  the  decimal  point  in  the 
quotient  directly  over  the  decimal  point  in  the  dividend  (or  under 
in  short  division^. 

When  the  divisor  contains  decimal  places^  make  a  dot  on  a  line 
with  the  tops  of  the  figures  as  many  places  at  the  right  of  the  deci- 
mal point  in  the  dividend  as  there  are  decimal  places  in  the 
divisor.  Place  the  decimal  point  in  the  quotient  directly  above 
this  dot  (or  below  in  short  division). 


MULTIPLICATION  AND  DIVISION  OF  DECIMALS       101 


Note  1.  If  there  is  a  remainder  after  all  the  figures  of  the  dividend 
have  been  used,  we  annex  ciphers  to  the  dividend,  and  continue  the  division 
until  there  is  no  remainder,  or  until  a  sufficient  number  of  decimal  places 
have  been  obtained  in  the  quotient. 

Note  2.  When  the  dividend  contains  fewer  decimal  places  than  the 
divisor,  we  annex  ciphers  to  the  dividend  until  it  has  as  many  decimal 
places  as  the  divisor. 


Written 

Find  the  quotients  and  test 

1.  60.8-^-1.6 

2.  .00075^.05 

3.  25. 50 -f-. 34 

4.  15.2 -3.04 

5.  .27560-^265 

6.  90.978-4-3.54 

7.  38.4444-4-177 

8.  14.4 -4-.  0018 

9.  1.127-4-4.9 

10.  .9156-4-12 

11.  315.432 -.48 

12.  1.5906 -4- .6 

13.  375 -4- .125  . 

14.  125 -4- .125 

15.  1000^.001 

16.  .001-4-1000 

17.  25-^.25 

18.  .25-4-25 


hi/  multiplication  : 

19.  18.65-4-100 

20.  266.4^.036 

21.  2.107 -4-. 35 

22.  .100854-4-3.879 

23.  125874.^.486 

24.  9801. -4-. 99 

25.  2976.^4.96 

26.  164. 32 --.208 

27.  347.76^.368 

28.  .0006478 -4-. 079 

29.  2.5826-5-69.8 

30.  98.07^.210 

31.  20.852 -4- .52 

32.  .0023322  -  .0026 

33.  676.8  -4-  .08 

34.  1273.998 -4-. 199 

35.  357.6-4-2.98 

36.  .75897-4-810 


102       MULTIPLICATION  AND  DIVISION  OF  DECIMALS 

147.  Find  the  products  and  test  hy  division  : 

1.  3.2x3.6  16.  7.0001  X. 0603 

2.  86  X  .09  17.  43.55  x  .06 

3.  9.8  X. 005  18.  2354  X. 008 

4.  .039x57  19.  39.04x2.08 

5.  .00356x6.8  20.  6.80x86732 

6.  6394.  X  .029  21.  2400.  x  .387 

7.  864.  X  .278  22.  .0406  x  3080 

8.  .00967x240  23.  920x63.7 

9.  839.42  X  .0015  24.  .992  x  3001 

10.  208.7x30.9  25.  2460  X. 0039 

11.  930x6.80  26.  1234.  x  .56 

12.  4203  X  .0076  27.  9.924  x  .0106 

13.  69  X. 00035  28.  .0204x20.40 

14.  406  X. 000039  29.  78.08  x  .025 

15.  7.92x1.002  30.  .8060x300 

148.  Oral 

.7      =  -j"^,  or  7  divided  by  10. 

.305  =  %%,  or  305  divided  by  1000. 

.581  =^ ,  or  58 J  divided  by  100. 

In  like  manner  tell  the  meanings  of  the  following  decimals  : 

1.  .8  6.    .89 J  11.    .029^  16.    .05 J 

2.  .416  7.    .6f  12.    .007^         17.    .034 

3.  .21  8.    .39f  13.    .103f  18.    .0165 

4.  .3879    9.  .48^    14.  .2134f    19.  .00017J 

5.  .200    10.  .873^    15.  .4070^    ao.  .OOOf 


CHANGING  DECIMALS  TO  FRACTIONS  103 

CHANGING   DECIMALS   TO   COMMON   FRACTIONS   OR   MIXED 

NUMBERS 


149. 

.072  = 

10  00  — 

ilr 

- 

18.25  =  18^^0  = 

=  18i 

Reduce  to 

common  fractions   or 

mixed 

numbers  in  simplest 

form : 

.1. 

.8 

8. 

.125 

15.   16.75 

2. 

.25 

9. 

.875 

16.    .00125 

3. 

.35 

10. 

.375 

17.    .054 

4. 

.75 

11. 

.455 

18.    .0250 

5. 

.64 

12. 

.025 

19.    .01375 

6. 

.52 

13. 

.561 

20.    .342 

7. 

.38 

14. 

.368 

12 
Solution :  .342  =  34f  ^  100  =  2^  X  yi-  =i?  Ans. 


21. 

.12J 

31. 

2.331 

5 

41. 

.166f 

22. 

.621 

32. 

'H 

42. 

.19^ 

23. 

.06} 

33. 

42.621 

43. 

.5621 

24. 

.18i 

34. 

97.087J 

44. 

400.40f- 

25. 

.03| 

35. 

56.131 

45. 

361.41-1 

26. 

.25f 

36. 

158.06} 

46. 

2042.  If 

27. 

.871 

37. 

409. 6| 

47. 

79.00f 

28. 

.66f 

38. 

•OTA 

48. 

308.00|| 

29. 

.361 

39. 

.261 

49. 

2890.901 

30. 

16.25 

40. 

.012f 

50. 

98.000|f 

104  KEDUCTION  OF  FRACTIONS  TO   DECIMALS    . 

150.  Oral 

Using  aliquot  parts^  reduce  the  following  to  mixed  numbers: 

1.  158.50  6.    16.05  11.    15.081 

2.  139.331  7.   I32.12J  12.   422.331 

3.  |5.16|  8.    $19.10  13.    100.14|- 

4.  17.25  9.    II8.I42  14.    603.16| 

5.  119.20  10.   16.02  15.    99.02 

151.  At  sights  reduce   the  following  to    common  fractions  or 
mixed  numbers : 


1. 

.621 

4.  12.875 

7.  29.871 

10.  43.4 

2. 

.66| 

5.  13.02 

8.  3.6 

11.  12.371 

3. 

.75 

6.  430.625 

9.  7.8 

12.  54.375 

REDUCTION    OF    COMMON    FRACTIONS   AND    MIXED    NUMBERS 

TO  DECIMALS 
152.    Written 

Change  -{^  to  a  decimal. 

.3125 

16)5.0000 
48 

"20 
16  ^g=  5 -5-16  =  . 3125  Ans,     . 

40  Write  9j-^g  as  a  mixed  decimal. 

32 

"so 

80 


REDUCTION   OF  FRACTIONS   TO   DECIMALS  105 


Ret 

iuce 

ifo  decimals 

.* 

1. 

t 

11. 

H 

21. 

*l 

31. 

19,^ 

2. 

1 

12. 

A 

22. 

If 

32. 

72T 

3. 

A 

13. 

i 

23. 

M 

33. 

llr 

4. 

i 

14. 

li 

24. 

H 

34. 

12^fT 

5. 

A 

15. 

3iV 

25. 

i¥ff 

35. 

14f| 

6. 

1 

16. 

f 

26. 

i¥ff 

36. 

StIt 

7. 

1 

17. 

^ 

27. 

12A 

37. 

ISiV 

8. 

3\ 

18. 

^r       . 

28. 

H 

38. 

2AV 

9. 

iV 

19. 

II 

29. 

If 

39. 

5A 

10. 

If 

20. 

2#j 

30. 

^2V 

40. 

3^ 

153.  ^  fraction  in -lowest  terms  whose  denomiyiator  contains 
other  prime  factors  than  2  arid  5  cannot  be  reduced  to  an  exact 
entire  decimal;  e.g.  |,  f,  if,  j\,  If,  ^\. 

Such  a  fraction  may  be  reduced  to  a  decimal  of  nearly  the 
same  value  by  carrying  the  division  to  a  certain  number  of  deci- 
mal places,  thus  : 

Reduce  ^|  to  a  decimal  of  four  places. 

,U01^^Ans. 
26.)19.0000 
182 

—  18=_9_     •'^307  is  almost  equalto  ||. 

^g         2  6      13      rp^g  gxact  value  of  if  is  .TSOT^V 


200 

182 

"Is 


The  result  may  be  expressed,  .7307-1- 


106  REDUCTION*  OF  FRACTIONS  TO  DECIMALS 

154.     Written 

Reduce  to  decimals  of  three  places: 


1- 1 

7. 

if 

13. 

8A 

19. 

42ii 

2.    ^ 

8. 

A 

14. 

231 

20. 

til 

3.    ^\ 

9. 

if 

15. 

681t\ 

21. 

2¥^ 

*•    f 

10. 

W 

16. 

Hy 

22. 

43A 

5.    f 

11. 

2^^ 

17. 

II 

23. 

16A 

6-    A 

12. 

151 

18. 

M 

24. 

in 

A  COMMON 

FRACTION 

AT  THE 

END  OF 

A  DECIMAL 

155.    .21  = 

.2+ a 

ofi^, 

or  2^.  or  , 

.05). 

.2 +  .05  = 

.25 

In  a  similar  manner,  we  may  show  that, 
.271  =  .275,  .3841  =  .3845,  etc. 

Also,  that  .21  =  .225,  .341  =  .3425,  etc. 

Also,  that  .8|  =  .875,  .06f  =  .0675,  etc. 

Also,  that  ,^  =  .9125,  .07|  =  .07375,  etc. 

Oral 

Express  as  entire  decimals: 


1.  a. 

'^ 

J.  $.17J 

c.  .3601 

d.  71 

e. 

.0041 

2.  a. 

'H 

5.  3.7^ 

c,  19. 20  J 

(^.  i.39} 

6. 

.145^ 

3.  a. 

.05f 

h.  l.lf 

c.  $21.46f 

d,  .033f 

e. 

.090f 

4.  a. 

•7i 

6.  6.81 

c.  80. 3 J 

(^.  12f 

e. 

1.9i 

5.  a. 

^2^ 

h,  3. 97 J 

c.  150f 

(?.  24.01 

e. 

29.0| 

6.  a. 

1.30f 

J.  2.45| 

<?.  lOOJ 

d.  20f 

e. 

40.001 

REVIEW  AND  PRACTICE  lOT 

REVIEW  AND  PRACTICE 
156.    Oral 

1.  Read : 

.0305;  42.0042;  4.0004;  6.395;  .0600;  639.500;  639.00005; 
10.10;   36f;  50.00|. 

2.  Express  as  common  fractions  in  lowest  terms: 

.5;   .25;   .75;    .121;    .37I;    .871;    .16|;    .33^;    .621;    .66|; 
.081;.  .02. 

3.  Divide  hi/  10  : 

62;  93.5;  3.44;  .25;  .0081;  .0384;  1.027. 

4.  Multiple/  ly  1000  : 

8.934;  .0245;  .612356;  48;  63.05. 

5.  Grive  results: 

436x100;  436 -i- 1000;  436x^10;  5-^100;  8314-^10,000; 
6.183  X  10,000. 

6.  Express  as  common  fractions  in  lowest  terms: 
.4;  .60;   .8;   .500;  .70;   .200;   .600. 

7.  How  may  decimals  be  divided  by  10  ?  by  100  ?  by  10,000  ? 
by  1,000,000?   by  1000? 

8.  A  stamp  clerk  received  $4085  for  special  delivery  stamps 
at  10  cents  apiece.     How  many  such  stamps  did  he  sell? 

9.  Compare  $1  with  |i  ;  If  with  If. 

10.  Compare  1  bushel  with  ^  of  a  bushel ;  5  bushels  with  | 
of  a  bushel. 

11.  What  will  3  months'  rent  cost  at  $  240  a  year  ? 

12.  What  will  a  dozen  cabbages  cost  at  the  rate  of  3  for  25^? 


108  REVIEW  AND  PRACTICE 


13.  a.  Count  the  eggs  in  the  top 
layer  of  this  crate.  How  many  dozen 
are  there  ? 

h.  How  many  such  layers  are  there  if 
the  crate  contains  30  dozen  ? 

c.  ^  of  them  are  how  many  dozen  ? 
How  many  eggs  ? 

d.  ^  of  one  layer  is  what  fraction  of  the  entire  number  ? 

e.  How  many  pounds  will  a  dozen  eggs  weigh  if  their  average 
weight  is  2  oz.  apiece  ? 

/.    What  is  the  entire  weight  of  the  eggs  in  the  crate  ? 

14.  If  I  of  a  yard  of  cloth  costs  50  cents,  what  will  3  yards 
cost  ?     (3  yd.  are  how  many  times  |  yd  ?) 

15.  If  I  borrow  one  dollar,  and  agree  to  pay  it  back  in  one 
year,  with  6  cents  more  to  pay  for  the  use  of  it,  how  much  must 
I  pay  in  all,  at  the  end  of  the  year  ? 

16.  If  you  lend  me  f  8  for  a  year,  and  I  agree  to  pay  five 
cents  for  the  use  of  each  dollar,  how  much  will  I  owe  you  at 
the  end  of  the  year? 

How  much  will  I  owe  if  I  agree  to  pay  5J  cents  for  the  use 
of  each  dollar?  7  cents?  8  cents?  4|^  cents?  6 J  cents? 
9  cents  ? 

17.  What  must  I  pay  for  the  use  of  1 20  for  one  year,  if  I  pay 
5  cents  for  the  use  of  each  dollar  ? 

If  I  keep  the  money  three  years,  paying  each  year  for  the  use 
of  it,  and  then  pay  back  the  $20  which  I  borrowed,  how  much 
will  I  have  paid  in  all  ? 

18.  Mr.  A  borrowed  9  dollars  from  Mr.  B.  After  four  years 
he  paid  it  back  and  also  paid  5  cents  a  year  for  the  use  of  each 
dollar.     How  much  did  Mr.  A  pay  to  Mr.  B  ? 


KEVIEW   AND  PRACTICE  109 

19.  A  man  borrowed  810  from  me  and  agreed  to  pay  me  for 
the  use  of  it  at  the  rate  of  6  cents  a  year  for  each  dollar.  He 
kept  the  money  only  half  a  year.  How  much  should  he  pay  me 
for  the  use  of  the  money  ? 

How  much  should  he  pay  me  in  all  ? 

20.  How  much  should  be  i)aid  for  the  use  of  twenty  dollars 
for  one  and  one  half  years  at  the  rate  of  5  cents  a  year  for  every 
dollar  ? 

21.  A  man  paid  $5  a  year  for  the  use  of  $100.  How  much 
a  year  would  he  pay  for  the  use  of  §500  at  the  same  rate  ? 

22.  If  $7  is  paid  for  the  use  of  a  sum  of  money  for  2  years, 
what  should  be  paid  for  the  use  of  the  same  sum  for  6  years  ? 

23.  A  woman  who  lives  near  the  grocery  takes  her  lamp  to 
the  grocery  to  be  filled  and  pays  5  cents  every  time.  The  lamp 
holds  a  pint  of  oil. 

a.    How  much  per  gallon  does  the  oil  cost  her  ?    , 
h.    If  the  lamp  burns  one  pint  of  oil  in  two  days,  in  how  many 
days  will  it  consume  one  gallon  of  oil  ? 

c.  In  how  many  days  will  it  consume  four  gallons  ? 

d.  How  much  would  she  save  by  buying  four  gallons  of  oil 
at  121^  a  gallon? 

24.  Margaret's  mother  buys  ^  lb.  of  sliced  bacon  twice  a 
week  at  23^  per  pound. 

a.  How  much  does  she  pay  each  time  ? 

h.   How  much  does  she  pay  for  bacon  in  six  weeks  ? 

c.  How  much  would  a  6-pound  piece  of  bacon  cost  at  20^  a 
pound  ? 

d.  How  much  would  Margaret's  mother  save  by  buying  such 
a  piece  at  that  price  ? 

25.  What  will  20  quarts  of  berries  cost  at  the  rate  of  2  quarts 
for  25)^? 


110  REVIEW  AND  PKACTlCfi 

157.     Written 

1.  Express  as  common  fractions  in  simplest  form:  a.  .125; 
b.  .625;  c.  .025;  d,  .5625;  e,  .087500;  /.  Five  hundred  five 
thousandths. 

2.  Express  as  exact  decimals: 

a.  ^;  h.   lif;  c.  ^g;  d.   gf o*'   ^-   ^q^;  /•   ^V 

3.  Reduce  to  decimals  of  four  places : 

1-    _3_7_.    IIA.      24_.    £5..    14. 
3»    111'     120'     1Y5'     91'    29- 

4.  Express  in  figures  and  add:  Six  and  forty-five  thou- 
sandths; forty-nine  and  seven  hundred  seventy-nine  thousandths; 
twenty -four  thousand  nine  hundred,  and  five  hundredths;  six 
hundred  five  thousandths;  three  hundred  forty-three  hundred- 
thousandths;  seventy-nine  and  five  tenths;  eleven  and  thirty- 
nine  hundredths;  eight  hundred  twenty-five,  and  forty-five 
hundredths;  nine  hundred  fifteen,  and  six  tenths. 

5.  From  fifteen  and  twenty-five  thousandths  take  five  and 
six  hundred  twenty-five  millionths.  Express  the  result  in 
words. 

6.  Find  the  sum  of  3.7;  36.05;  .085;  35.25. 

7.  From  .8  take  eight  thousandths. 

8.  Find  the  sum  of  .0375,  241,  43.8,  and  7-|. 

9.  Find  the  cost  of  Q,^  barrels  of  flour  at  $5.75  a  barrel. 

10.  What  number  divided  by  17|  will  give  5.05? 

11.  Add  36.048,  25.13,  13|,  and  25|. 

12.  Find  the  sum  of  |  and  .375. 

13.  A  liveryman  bought  10.5  tons  of  hay  at  $10,375  a  ton. 
What  did  it  cost? 

14.  At  $60.75  an  acre,  what  is  the  cost  of  40.25  acres  of 
land? 


REVIEW  AND  PRACTICE 


111 


15.  A  lady  purchased  4.5  yd.  of  silk  at  $1.25  per  yard  and 
7.25  yd.  of  broadcloth  at  $3.50  per  yard.  What  change  should 
she  receive  from  a  $  20  bill  and  two  $  10  bills  ? 

16.  A  farmer  divided  his  farm  of  168.8  acres  into  16  equal 
fields.     How  much  land  was  there  in  each  field? 

17.  Divide  53.66523  by  .941. 

18.  28.8-^.0072  =  ? 

19.  Divide  .305996  by  .337. 

20.  Divide  990  by  .11. 

21.  $87.50  will  pay  for  how  many  piano  lessons  at  $1.25 
a  lesson? 

22.  207.09 


,015  =  ? 


23.  A  boy's  height  increased  1.4  inches  in  his  12th  year,  1.5 
inches  in  his  13th  year,  and  1.6  inches  in  his  14th  year.  What 
was  his  average  growth  per  year  for  the  three  years? 

24.  A  farmer  received  $146^  for  pigs  at  $3J  apiece.  How 
many  pigs  did  he  sell? 

25.  The  smaller  of  two  numbers  is  9346.05.  Their  differ- 
ence is  412.08.     What  is  the  greater? 

26.  9.2x5.37 -(1.785 +  .1318)  =  ? 

27.  The  following  quantities  of  stamps  were  sold  during  a 
year  at  a  city  post  office.  Find  what  was  received  for  all  of 
them  : 

1-cent  .  .  .  4,832,500  6-cent  .  .  .  40,000 

2-cent  .  .  .  10,550,000  8-cent  .  .  .  47,100 

3-cent  .  .   .  117,500  10-cent  .  .  .  70,200 

4-cent  .  .  .  161,000  13-cent  .  .  .      3400 

5-cent  .  .  .  198,000  15-cent  .  .  .      8800 


112 


REVIEW  AND  PKACTICE 


PART  OF  AN  AMERICAN  WAR  FLEET 
Battleships 


NAME 

LENGTH 

TONNAGE 

NO.   OF  GUNS 

SPEED  IN  KNOTS 
PER  HOUR 

MEN 

Maine  ..... 
Missouri  .... 

388  ft. 
388  ft. 

13500 

14278 

44 

44 

18 
18.2 

648 
652 

Kentucky  .  .  . 
Kearsarge  .  .  . 
Louisiana  .  .  . 

368  ft. 
368  ft. 
450  ft. 

12905 
12905 
17666 

60 
56 
74 

16.9 
16.8 

18.8 

613 
650 

803 

Rhode  Island  . 

435  ft. 

16125 

66 

19.5 

812 

New  Jersey  .  . 
Virginia  .... 
Alabama.  .  .  . 

435  ft. 
435  ft. 
368  ft. 

16125 
16125 
12170 

66 
66 

48 

19.5 
19.5 
17 

812 
812 
592 

Illinois 

368  ft. 

12170 

46 

17.5 

660 

Indiana  .... 

348  ft. 

11403 

45 

15.5 

491 

Iowa 

360  ft. 

12445 

48 

17.1 

520 

Armored  Cruisers 


West  Virginia. 

502  ft. 

13680 

60 

22.2 

648 

Pennsylvania  . 

502  ft. 

13780 

60 

22.9 

750 

Colorado.  .  .  . 

502  ft. 

13780 

60 

22.2 

750 

Maryland  .  .  . 

502  ft. 

13680 

60 

22.4 

.648 

28.    a.    If  all  these  ships  were  formed  in  a  single  unbroken 
line,  how  many  feet  long  would  the  line  be  ? 
h.    How  many  miles  long  would  it  be  ? 

c.  What  is  the  average  tonnage  ? 

d.  How  many  guns  do  they  all  carry  ? 

e.  How  many  men  are  there  on  all  the  ships  ? 

/.    How  many  knots  greater  is  the  average  speed  of  the 
cruisers  than  of  the  battleships? 
g.    What  is  the  average  number  of  men  on  a  ship  ? 


REVIEW  AND  PRACTICE 


113 


The  Battleship  Louisiana 


29.  A  knot,  or  nautical  mile,  is  used  in  measuring  the  speed 
of  vessels.     It  is  equal  to  about  1.15  common  or  statute  miles. 

a.  The  Missouri's  speed  is  how  many  common  miles  per 
hour? 

h.  The  Maryland's  speed  per  hour  is  how  many  common 
miles  greater  than  the  Missouri' s? 

30.  a.  A  knot  is  6080.27  ft.  How  many  feet  farther  can 
the  cruiser  Maryland  sail  in  an  hour  than  the  battleship 
Alabama  f 

h.  If  the  Alabama  is  9.9  knots  ahead  of  the  Louisiana^  and 
both  are  moving  at  highest  speed,  in  how  many  hours  can  the 
Louisiana  overtake  the  Alabama? 

31.  How  many  common  miles  can  the  'Kentucky  go  in  24 
hours,  if  she  goes  16.9  knots  per  hour  all  the  time  ? 

32.  How  many  feet  can  the  Rhode  Island  go  in  1  minute  ? 

Make  other  problems  about  these  warships,  using  the  num- 
bers given  in  the  table. 


114  REVIEW  AND  PRACTICE 

33.  What  is  the  cost  of  8.2  bales  of  cotton,  each  bale  weigh- 
ing 412.6  lb.  at  I J  a  pound  ? 

34.  Two  motor  cars  start  from  the  same  place  at  the  same 
time  and  go  in  opposite  directions,  one  at  the  rate  of  12.325 
miles  an  hour,  and  the  other  at  the  rate  of  14.875  miles  an 
hour.     How  far  apart  are  they  at  the  end  of  10|  hours  ? 

35.  a.  What  must  be  paid  for  the  use  of  $225  for  one  year 
at  the  rate  of  6  cents  a  year  for  the  use  of  one  dollar  ?  h.  For 
two  years  ?     c.    For  3|  years  ? 

36.  Mr.  Scott  borrowed  il500  from  Mr.  Moore  and  agreed 
to  pay  it  back  in  two  years,  and  also  to  pay  Mr.  Moore  for  the 
use  of  it  at  the  rate  of  5  cents  a  year  for  every  dollar. 
a.  How  much  did  Mr.  Scott  have  to  pay  for  the  use  of  the 
money  ? 

5.   How  much  did  he  have  to  pay  in  all  ? 

37.  If  I  pay  42  dollars  for  the  use  of  $600  for  a  year,  how 
much  do  I  pay  for  the  use  of  one  dollar  ? 

38.  Find  how  much  I  will  pay  in  2-|-  years  for  the  use  of  $  2000 
if  I  pay  4|-  cents  each  year  for  the  use  of  every  dollar. 

39.  Mr.  Marvin  borrowed  $3500  and  paid  it  five  years  after- 
ward. He  also  paid  4  cents  each  year  on  every  dollar  he  owed. 
How  much  did  he  pay  in  all  ? 

40.  If  I  pay  at  the  rate  of  T  cents  a  year  for  the  use  of  one 
dollar,  how  much  must  I  pay  for  the  use  of  $2800  for  six 
months?     (Six  months  are  what  part  of  a  year?) 

41.  a.  If  5  cents  will  pay  for  the  use  of  one  dollar  for  a  year, 
$24  will  pay  for  the  use  of  how  many  dollars  for  a  year  ? 

J.  At  the  rate  of  6  cents  a  year  for  the  use  of  a  dollar,  $24 
will  pay  for  the  use  of  how  many  dollars  for  a  year  ? 


REVIEW  AND  PRACTICE 


115 


^ 

|piBS 

HiJ^^^^HBflHH^ 

2*  -^ 

3j^ 

H|^^^H  \ 

^ 

^a| 

J/i 

1    ' 

^^M 

^L 

42.  These  loaves  of  bread  are  11^  inches  long.  The  blade 
of  the  bread  cutter  revolves  so  as  to  cut  off  a  slice  of  bread  at 
every    turn     of    the 

wheel. 

a.  If  the  slices  are 
■^^  in.  thick,  how 
many  inches  of  the 
loaf  will  be  cut  off 
by  18  turns  of  the 
wheel  ? 

h.  How  many 
inches  will  be  left  ? 

e.  How  many 
turns  are  made  in 
cutting  an  entire 
loaf  ? 

£?.   How  many  slices  are  made  from  one  loaf  ? 

43.  A  loaf  of  this  bread  is  made  into  sandwiches.  The 
slices  are  y^g  in.  thick.  The  crusts  are  not  used  for  this  purpose. 
Each  sandwich  is  made  of  two  slices  of  bread  and  one  ounce  of 
meat  that  costs  16^  a  pound.     The  bread  costs  5/  a  loaf. 

a.    How  many  sandwiches  are  made  ? 
h.    What  is  their  entire  cost  ? 

c.  If  they  are  sold  at  5  ^  apiece,  what  is  the  profit  on  all  of 
them  ? 

44.  There  are  90  loaves  of  bread  in  the  picture. 

a.  If  they  are  11^  in.  long  and  cut  into  slices  |-  in.  thick, 
how  many  slices  are  there  ? 

h.  If  the  bread  costs  5j^  a  loaf  and  all  except  the  crusts  is 
served  at  a  lunch  counter  at  1  ^  a  slice,  what  is  the  profit  on  all 
of  it? 

c.  The  90  loaves  would  make  how  many  slices  f  in.  thick? 


116       MULTIPLYING  AND  DIVIDING  BY  TWENTY-FIVE 

45.  A  barrel  of  flour  weighs  196  lb.  Flour  costing  i5  a 
barrel  is  made  into  bread  containing  llj  oz.  of  flour  to  a  loaf. 
Some  of  the  bread  is  bought  by  Mrs.  X  at  5^  a  loaf. 

a.  How  many  loaves  of  bread  are  made  from  a  barrel  of 
flour  ? 

h.    How  much  does  it  bring  at  5^  a  loaf  ? 

c.  How  much  would  Mrs.  X  save  by  buying  a  barrel  of 
flour  and  making  her  bread,  supposing  that  the  yeast  costs 
50^  and  the  extra  coal  80^? 

MULTIPLYING  AND  DIVIDING  BY   TWENTY-FIVE 

158.  25  =  100  -i-  4.  Therefore  we  may  multiply  a  number  hy 
25  hy  multiplying  it  hy  100  and  dividing  the  product  hy  4,  thus : 
36  X  25  =  3600  ^  4  =  900. 

How  did  we  multiply  36  by  100  ? 

159.  Oral 

Multiply  the  following  numhers  hy  25: 
12,  24,  16,  20,  48,  36,  44,  32,  8,  28,  40. 


1.  Multiply  by  25 : 

a.    63 

e.    76.347 

i. 

1124 

m. 

2.463 

h.    7428 

/.  14.231 

J- 

9.370 

n. 

2468  tons 

c.    231 

g.   2.31 

k. 

21.3 

0. 

.0934 

d.   155 

h.  .2835 

h 

84.8 

P- 

.00080 

2.    Find  the  cost  of  : 

a.  97  stoves  at  $25  apiece. 

h.  25  yd.  of  silk  at  $1.39  a  yard. 

c.  130  acres  of  land  at  $25  an  acre. 

d.  A  12-pound  turkey  at  25  cents  a  pound. 


ACCOUNTS  AND  BILLS  117 

160.    To  divide  a  number  hy  25  we  may  point  off  two  decimal 
'places  in  the  number  and  multiply  the  result  by  4,  thus  : 

452  -^  25  =  4.52  x  4  =  18.08 

^  4.52 

Explanation ;  452  -^  25  =  ^  -i-  i^  =  ^  X  ^  =  18.08 
^  1         4         1       100 


37.5 -^  25  =  .375  x  4  =  1.500 

.375 


Explanation:  2:1, b-^2b=  2^1. b-^^  =  ^x,^  =  lMO 
^  4  1        100 


161.     Written 

1.    Divide 

by 

25: 

a.    425 

e.    6934 

{.  63,940 

m.  $348.5 

b.    37.8 

/.  5876 

y.  6234 

w.  12964 

c.    239 

g,   93.6 

k.  $934 

0.   832  pk. 

d.    87.64 

h.   98,301 

Z.  150.25 

^.  .637 

2.    a.  831 

-f- 

25  =  ?        b. 

6.934 

X  25  =  ? 

c.  25  X  ?  =  .21 

3.    a.  428 

= 

25  X  ?        6. 

389 -T 

-  ?  =  25. 

^.  ?  X  25  =  12.6 

ACCOUNTS  AND  BILLS 

162.  When  your  mother  sends  you  to  the  store  where  she  is 
accustomed  to  buy  groceries,  giving  you  no  money  to  take  with 
you,  and  tells  you  to  buy  certain  articles  and  have  them 
charged,  what  does  she  mean  ? 

The  merchant  has  a  book  in  which  he  keeps  the  names  of 
persons  to  whom  he  sells  things  not  paid  for  at  the  time  of  the 
sale,  together  with  a  list  of  the  articles  sold,  their  value,  and 
the  date  of  sale.     This  list  is  called  an  account. 


118  ACCOUNTS  AND  BILLS 

The  person  who  sells  the  goods  is  the  creditor,  and  the  person 
who  buys  the  goods  is  the  debtor.  The  debtor  and  creditor  are 
called  parties  to  the  account. 

A  doctor  keeps  a  record  of  the  calls  which  he  makes  or  re- 
ceives in  treating  his  patients,  when  the  calls  are  not  paid  for 
at  the  time.  This  record  is  called  the  patient's  account.  Who 
is  the  creditor  ?     Who  is  the  debtor  ? 

If  your  father  works  for  some  one,  he  keeps  an  account  of  his 
time  and  wages.     Which  party  is  your  father  ? 

Whenever  something  is  paid  toward  a  debt  of  this  kind,  a 
record  of  the  payment  is  put  in  the  account  and  is  called  a 
credit.  The  difference  between  the  amount  of  the  debit  (or 
owing)  items  and  the  amount  of  the  credit  items  is  called  the 
balance  of  the  account. 

At  certain  times,  the  creditor  copies  on  a  piece  of  paper  a 
statement  of  the  debtor's  account  and  sends  it  to  the  debtor. 
This  statement  is  called  a  bill.  Some  merchants  always  send  a 
bill  with  the  goods  at  the  time  of  purchase. 

163.  There  are  various  ways  of  writing  a  bill,  but  it  should 
always  contain  these  things: 

1.  The  time  and  place  of  making  out  the  bill.  This  is  called 
the  date  of  the  bill. 

2.  The  debtor's  name  and  address. 

3.  The  creditor's  name  and  address. 

4.  A  list  of  the  items — that  is,  the  goods  sold,  money  paid 
or  services  rendered,  with  the  amount  of  each  item. 

5.  The  date  of  each  transaction,  if  any  of  them  occur  at  any 
other  time  than  that  of  making  out  the  bill. 

6.  The  amount,  or  footing,  of  the  bill. 


BILLS 


119 


164. 


FORMS  OF  BILLS 
(Form  i) 

Syracusb,  N.  Y.,  June  20,  1907. 


Mr.  John  P.  Smith, 

713  McBride  St., 


Bought  of  Andrews  Brothers, 

cor.  James  and  Warren  Sts. 


2  bu.  Apples 


$1.10 


10  lb.  Granulated  Sugar    .05| 


I  lb.  Tea 


.50 


12 


20 
55 
25 


$3 


00 


What  is  the  date  of  this  bill? 

Who  is  the  creditor?     The  debtor? 

Read  the  items. 

What  is  the  amount  of  the  bill? 

Who  ought  to  pay  the  bill? 

Who  should  receive  the  money? 

When  a  bill  is  paid,  the  creditor  "receipts"  the  bill  by 
writing  at  the  bottom,  "Received  Payment,"  followed  by  the 
date  and  his  own  name.  This  shows  that  the  bill  has  been 
paid.  The  debtor  keeps  the  receipted  bill  to  show  that  the 
debt  has  been  paid. 

When  the  above  bill  is  paid,  who  should  receipt  it  ? 

Make  a  bill  similar  to  the  one  given  above,  but  using  different 
items.     Find  its  amount,  and  receipt  it. 


120 


BILLS 


(Form  2) 


Mr.  W.  C.  Flint, 
1213  Maxwell  Ave., 

To  John  M.  Semple,  M.D.,  J)r. 
207-208-209  Jamieson  Building 


Spokane,  Wash., 
April  1,  1907. 


To  professional  services  rendered,  Feb.  25  to  March 

18,1907,  $37  50 

Received  Payment, 
April  16,  1907. 
John  M.  Semple. 

Name  each  of  the  parties  in  this  bill.  Has  the  bill  been 
paid?     How  do  you  know? 

Sometimes  a  clerk,  an  agent,  or  a  bookkeeper  of  the  creditor 
receives  the  money  for  payment  of  a  bill.  He  should  then 
write  the  creditor's  name  under  the  words  "  Received  Pay- 
ment," and  under  the  creditor's  name,  his  own  name  or  initials, 
thus  : 

Received  Payment, 

John  M.  Semple, 

Per  Kate  L.  Bunn. 

3.  Receipt  the  bill  in  Form  1  as  though  you  were  a  clerk  for 
the  creditor. 

4.  Make  out  bills  of  the  following  items,  the  teacher  being 
the  debtor  in  each  one,  and  yourself  the  creditor.  Let  the 
teacher  examine  each  bill,  and  mark  it  O.K.  if  correctly  made. 
Receipt  the  bill  and  return  it  to  the  teacher. 


VOLUME  MEASURE  121 

a.    3  bu.  potatoes  at  75i^  per  bushel. 

8  lb.  lard  at  15^  per  pound. 

5  gal.  kerosene  oil  at  12/  per  gallon. 

h,   4  lb.  coffee  at  25/  per  pound. 
18  lb.  sugar  at  5|-/  per  pound. 

5  gal.  molasses  at  60/  per  gallon. 

c.  6  bbl.  potatoes  at  $1.80  per  barrel. 

2  tons  hay  at  §16  per  ton. 

3  cords  wood  at  f  4  per  cord. 

d.  16  yd.  silk  at  11.50  per  yard. 

4  pairs  hose  at  50  /  per  pair. 

9  yd.  lace  at  60/  per  yard. 

e.  1  chocolate  pot,  75/. 

6  salad  plates  at  $1.10  each. 

15  Haviland  bouillon  cups  at  $12  per  doz. 
1  bread  plate,  98/. 

VOLUME  MEASURE 

165.  Anything  that  has  lengthy  breadth^  and  thickness  is  a  solid ; 
as  wood,  stone,  earth. 

166.  A  solid  hounded  hy  six  square  faces  is  a  cube. 

167.  A  solid  hounded  hy  six  rectangles  is  a  rectangular  prism. 
Name  as  many  objects  as  you  can  that  are  rectangular  prisms. 

168.  The  lengthy  hreadth^  and  thickness  of  a  solid  are  its  dimen^ 
sions. 

169.  A  cube  whose  edge  is  1  inch  is  a  cubic  inch. 

Show  with  your  hands  how  long,  wide,  and  high  a  cubia 
inch  is. 


122  VOLUME  MEASURE 

170.  A  cube  whose  edge  is  1  foot  is  a  cubic  foot. 

Show  with  your  hands  the  length,  breadth,  and  height  of  a 
cubic  foot. 

Show  how  high  a  cubic  foot  would  be  if  it  were  lying  on 
your  desk. 

Can  you  think  of  some  object  about  as  large  as  a  cubic  foot  ? 

171.  A  cube  whose  edge  is  1  yard  is  a  cubic  yard. 

Show  with  your  hands  how  wide  and  high  a  cubic  yard  is. 
Show  how  high  it  would  reach  if  it  stood  on  the  floor  by  your 
side. 

Could  a  cubic  yard  be  put  through  the  open  door  or  window 
of  your  school-room  ?     Measure  and  see. 

172.  The  number  of  cubic  yards^  cubic  feet^  or  cubic  inches  that 
a  solid  contains  is  its  contents  or  volume. 

Name  some  object  whose  volume  is  about  1  cubic  yard. 

The  measure  by  which  the  contents  of  a  solid  are  measured  is 
called  volume  measure. 

To  THE  Teacher. — Cubes  of  various  sizes  should  be  provided  for  this 
lesson.  Inch  cubes  should  be  put  together  to  make  two-inch  cubes,  three- 
inch  cubes,  and  so  on. 

The  teacher  should  gain  from  children  that  each  block  has  six  equal 
square  faces,  and  therefore  is  a  cube.  Make  prisms  with  the  cubes,  and 
gain  from  the  children  that  each  face  of  the  prism  is  a  rectangle ;  also  that 
the  volume  of  a  rectangular  prism  is  the  product  of  its  three  dimensions. 

A  piece  of  board  one  foot  square  and  one  inch  thick  can  be  used  effect- 
ively. Mark  it  off,  to  show  that  it  contains  144  cubic  inches.  Gain  from 
children  that  12  such  boards,  piled  one  upon  another,  would  make  a  cubic 
foot.  Have  children  draw  patterns  of  inch  cubes,  cut,  and  paste  them. 
Draw  on  the  blackboard  a  pattern  of  a  cubic  foot. 

Work  with  such  exercises  as  are  here  outlined  until  pupils  are  perfectly 
familiar  with  these  fundamental  ideas  of  volume  measure.  Do  not  take  for 
granted  that  children  have  these  ideas,  but  test  their  knowledge  by  requir- 
ing them  to  show  and  construct. 


VOLUME  MEASURE 


123 


[  * 

/                                             \ 
\ / 

V 

This  is  a  pattern  of  a  cubic  inch,  or  an  inch  cube.  When  it  is 
drawn  on  paper,  an  inch  cube  may  be  made  by  cutting  along 
the  dark  lines,  folding  along  the  light  lines,  and  pasting  the 
flaps  over  to  hold  the  edges  together. 

How  many  faces  has  an  inch  cube?  What  is  the  size  of 
each  face  ? 

Draw  a  pattern  of  a  cubic  inch.  Cut  it  out  and  paste  it  so 
as  to  make  a  cubic  inch. 

A  foot  cube  has  how  many  faces  V  What  is  the  size  of  each 
face? 

On  a  large  sheet  of  stiff  paper,  at  home,  draw  a  pattern  of  a 
cubic  foot.     Bring  it  to  school  and  paste  it  together. 


124 


VOLUME  MEASURE 


VnXVv 


X 


\,  \.  \,  \. 


\ 


1.  Using  inch  cubes,  make  a  rectan- 
gular prism,  4  in.  by  5  in.  by  1  in. 
How  many  cubic  inches  does  it  contain? 
5x4x1=? 

2.  If  three  such  prisms  as  figure  A 
were  piled  up,  they  would  make  a  prism 
containing  how  many  cubic  inches  ? 
5x4x3=? 

3.  How  many  cubic  feet  are  there 
in  a  rectangular  prism  5  ft.  by  4  ft.  by 
3  ft.  ? 


4.    How  many  cubic  yards  are  there 
in  a  rectangular  prism  5  yd.  by  4  yd.  by  3  yd.  ? 

5.  A  rectangular  prism  6  in.  long,  3  in.  wide,  and  2  in.  high 
contains  how  many  cubic  inches  ? 

Make  this  prism  from  paper  and  find  the  area  of  each  face 
and  all  its  faces. 

6.  Find  the  contents  of  a  2-inch  cube.    Find  its  entire  sur- 
face. 

7.  A  dry-goods  box  is  6  ft.  long,  8  ft.  wide,  and  3  ft.  deep. 
How  many  cubic  feet  of  space  will  it  occupy  in  a  freight  car  ? 

8.  Find  the  contents  of  a  4-inch  cube.     Find  its  entire  sur- 
face. 

9.  A  block  8  inches  long,  2  inches  wide,  and  2  inches  high 
will  make  how  many  2-inch  cubes  ? 

10.  From  a  piece  of  paper  8  in.  square,  Mabel  cut  enough  to 
cover  a  box  2  in.  by  4  in.  by  1  in.  How  many  square  inches 
of  paper  were  used?     How  many  were  left  ? 


VOLUME  MEASURE  125 

11.  How  many  cubic  inches  in  a  cube  of  soap  4  in.  by  3  in. 
by  11  in.? 

12.  If  a  pasteboard  box  is  4  in.  square,  how  high  must  it  be 
to  contain  32  cu.  in.  ?     4  x  4  x  ?  =  32.     16  x  ?  =  32. 

13.  A  stick  of  wood  is  3  in.  wide  and  2  in.  thick.  How  long 
must  it  be  to  contain  60  cu.  in.  ?     3  X  2  x  ?  =  60.     6  x  ?  =  60« 

14.  2  X  5  X  ?  =  70.     ?  X  3  X  4  =  48.     7  x  ?  x  2  =  28. 

15.  If  a  cubic  foot  of  ice  weighs  60  lb.,  what  is  the  weight  of 
a  cake  of  ice  2  ft.  by  1  ft.  by  1  ft.  ? 

16.  Will's  lunch  box  is  8  in.  long,  4  in.  wide,  and  3  in.  thick. 
What  is  its  volume  ? 

17.  If  a  common  brick  is  8  in.  by  4  in.  by  2  in.,  how  many 
cubic  inches  does  it  contain  ?  What  is  the  area  of  one  of  its 
largest  faces?  How  many  such  faces  has  it?  What  is  the 
area  of  one  of  its  smallest  faces  ?  How  many  such  faces  ? 
What  is  the  area  of  the  other  faces  ? 

Note.  —  Bring  a  brick  to  the  class  and  verify  these  results. 

18.  Find  the  contents  of  a  rectangular  prism,  3  in.  by  4  in. 
by  21  in. 

19.  A  cubic  foot  is  how  many  inches  wide  ?  High  ?  Long  ? 
It  contains  how  many  cubic  inches  ?  Can  you  show  this  by  a 
drawing  ? 

20.  A  cubic  yard  is  how  many  inches  wide,  high,  and  long  ? 
It  contains  how  many  cubic  inches  ?     Show  this  by  a  drawing. 

21.  How  many  square  inches  in  one  face  of  a  cubic  foot  ? 

22.  How  many  square  feet  in  one  face  of  a  cubic  yard  ?  In 
all  its  faces  ? 

23.  Learn  the  table  of  Volume  Measure,  p.  173. 


126  VOLUME  MEASURE 

174.    Written 

1.  How  many  cubic  inches  are  there  in  one  cubic  yard  ? 

2.  a.  How  many  cubic  inches  are  there  in  5|  cubic  feet  ? 
b.  In  lOf  cubic  feet  ?  c.  In  25^  cubic  feet  ?  d.  In  307  cubic 
feet? 

3.  a.  How  many  cubic  feet  are  there  in  70|^  cubic  yards  ? 
b.  In  235|^  cubic  yards  ?  c.  In  .16|  of  a  cubic  yard?  d.  In 
21.33^  cubic  yards  ?     e.    In  4.66|  cubic  yards  ? 

4.  9234  cu.  ft.  =  how  many  cubic  yards  ? 

5.  How  many  cubic  feet  are  there  in  24,192  cubic  inches  ? 

6.  How  many  cubic  feet  are  equal  to  84  cu.  yd.  17  cu.  ft.? 

7.  Change  38  cu.  ft.  347  cu.  in.  to  cubic  inches. 

8.  A  cake  of  ice  containing  S^  cu.  ft.  contains  how  many 
cubic  inches  ? 

9.  How  many  cubic  feet  are  there  in  a  bowlder  that  contains 
7|  cu.  yd.  ? 

10.  Change  13  cu.  yd. ;  a.  to  cubic  feet ;  b.  to  cubic  inches. 

11.  Change  513,216  cu.  in.  to  cubic  feet. 

12.  A  carload  of  earth  containing  19.8  cu.  yd.  contains  how 
many  cubic  feet  ? 

13.  7  X  8  X  ?  =  280.  The  contents  of  a  rectangular  prism 
are  280  cu.  in.  Two  of  its  dimensions  are  7  in.  and  8  in. 
What  is  the  other  dimension  ? 

14.  The  dimensions  of  a  water  tank  are  10  ft.,  11  ft.  and 
2 J  ft.     What  is  the  volume  of  the  tank? 

15.  A  gallon  contains  231  cubic  inches.  Seven  gallons  are 
how  many  cubic  inches  less  than  one  cubic  foot  ? 


VOLUME  MEASURE  127 

16.  A  pile  of  wood  8  ft.  long,  4  ft.  wide,  and  4  ft.  high  con- 
tains how  many  cubic  feet  ? 

17.  A  stick  of  timber  8  in.  square  and  45  ft.  long  contains 
how  many  cubic  inches  ?     (First  change  45  ft.  to  inches.) 

18.  a.  A  block  of  granite  16  ft.  by  3J  ft.  by  2|  ft.  contains 
how  many  cubic  feet  ?     h.    How  many  cubic  inches  ? 

19.  A  flagstone  8  ft.  long,  5  ft.  wide,  and  6  in.  thick  contains 
how  many  cubic  feet  ? 

Hint  :   The  stone  is  what  part  of  a  foot  in  thickness  ? 

20.  A  street  100  rods  long  must  be  lowered  3  ft.  in  order 
that  a  pavement  36  ft.  wide  may  be  laid.  a.  How  many  yards 
long  is  the  street?      h.    The  cut  is  how  many  yards  wide? 

c.  How  many  cubic  yards  must  be  cut  out  ? 

21.  a.  A  box  car  30  ft.  long,  9  ft.  wide,  and  7  ft.  high  con- 
tains how  many  cubic  feet  of  space  ?  h.  How  many  cubic 
yards  ? 

22.  A  school-room  is  39  ft.  long,  30  ft.  wide,  and  12  ft.  high. 
a.  How  many  cubic  feet  of  air  space  are  there  in  the  room? 
h.  How  many  cubic  yards  ?  c.  If  there  are  40  pupils  in  the 
room,  how  many  cubic  feet  of  space  are  there  for  each  pupil  ? 

d.  How  many  cubic  yards  are  there  for  each  pupil  ? 

175.    Oral 

1.  What  three  numbers  multiplied  together  equal  12  ? 

2.  What  dimensions  could  a  cake  of  maple  sugar  have  to 
contain  12  cubic  inches  ? 

3.  4  X  3  X  ?  =  24.     A  block  of  wood  4  in.  long  and  3  in. 
wide  must  be  how  thick  to  contain  24  cubic  inches  ? 

4.  What  dimensions  could  a  box  have  to  hold  36  cu.  ft.  of 
coal  ? 


128  VOLUME  MEASURE 

5.  What  dimensions  could  a  box  have  to  hold : 

a.   40  inch-cubes  ?         h.    18  inch-cubes  ?         c,   32  inch-cubes  ? 

6.  A  box  is  4  in.  wide  and  IJ  in.  deep.  It  contains  72  cubic 
inches  ?     How  long  is  it  ? 

7.  How  many  inch-cubes  are  equal  to  a  rectangular  prism 
3  in.  by  2  in.  by  5  in.  ? 

176.    Written 

1.  A  man  engaged  to  dig  a  cellar  40  ft.  long,  25  ft.  wide, 
and  6  ft.  deep.  How  much  has  he  to  dig  after  he  has  taken  out 
3246  cu.  ft.  of  earth  ? 

2.  How  many  bricks  4  in.  by  8  in.  by  2  in.  are  equal  to  a 
cubic  foot  ? 

3.  Henry  gathered  2  boxes  full  of  hickory  nuts.  The  boxes 
measured  6  in.  by  14  in.  by  10  in.  and  7  in.  by  9  in.  by  12  in. 
He  emptied  the  nuts  into  a  box  20  in.  long,  18  in.  wide,  and  1 
ft.  deep.     How  much  space  was  left  in  the  large  box  ? 

4.  a.  A  tank  2  ft.  by  3  ft.  by  4  ft.  contains  how  many  cubic 
feet?  6.  How  many  cubic  inches  ?  c.  If  there  are  100  gallons 
of  water  in  the  tank,  how  many  cubic  inches  of  water  are  there  ? 
d.    How  many  more  cubic  inches  of  water  will  fill  the  tank  ? 

5.  If  a  box  of  candy  1|"  by  4''  by  6"  is  sold  for  19  cents, 
what  should  be  the  price  of  a  box  6"  by  3''  by  6"  filled  with 
the  same  kind  of  candy  ? 

6.  a.  How  many  cubic  feet  of  earth  are  required  to  fill  an 
old  cellar  30  ft.  by  90  ft.  and  7^  ft.  deep  ? 

h.  How  many  loads  of  earth  are  required  if  one  load  will  fill 
IJ  cu.  yd.  of  space  ? 

7.  Make  and  solve  a  problem  about  the  volume  of  a  wagon  box. 

8.  Make  and  solve  a  problem  about  the  quantity  of  air  in  a 
room. 


PART   TWO 

To  THE  Teacher. '-It  will  be  found  worth  while  to  give  frequent  "quick 
tests,"  specimens  of  which  are  given  in  this  and  the  primary  book.  Dic- 
tate the  questions  one  at  a  time,  and  allow  a  few  seconds  (more  or  less  time 
according  to  the  nature  of  the  question)  for  the  pupils  to  obtain  the  answer 
mentally.  Then  at  a  given  signal  let  all  write  the  result  at  the  same  time. 
Allow  no  use  of  pen  or  pencil  except  to  write  results.  This  exercise  will 
cultivate  power  of  attention,  concentration,  and  alertness.  Let  it  be  short, 
sharp,  and  rapid  enough  to  keep  pupils  doing  their  best. 


REVIEW  AND  PRACTICE 
177.    Oral 

1.  $8  will  buy  how  many  pounds  of  butter  at  25^  a  pound? 

2.  At  33 J  ^  a  bushel,  13  will  pay  for  how  many  bushels  of 
turnips  ? 

3.  $1  is  the  cost  of  how  much  cheese  cloth  at  6|/  a  yard? 

4.  Find  the  cost  of  24  yd.  of  ribbon  at  8J^  a  yard. 

5.  When  $4  will  buy  24  lb.  of  beefsteak,  how  many  pounds 
will  il  buy?     How  much  does  a  pound  cost? 

6.  When  i9  will  pay  for  72  cans  of  peas,  what  is  the  price 
per  can?     (How  many  cans  for  a  dollar?) 

7.  How  long  will  $6.00  worth  of  stamps  last  a  man  who 
uses  1.30  worth  every  day? 

8.  Find  the  cost  of  300  yd.  of  flannel  at  33j/  a  yard. 

129 


130  REVIEW  AND  PRACTICE 

9.    Grive  the  products  : 
|x2;lx3;2xll;|x3;lxl;lx|;^x|;ix|. 

10.  2  qt.  of  beans  at  TJ^  and  3  lb.  of  maple  sugar  at  S^^cost 
how  much? 

11.  How  many  one-ounce  samples  can  be  made  from  2^  lb. 
of  cereal? 

12.  How  many  yards  are  there  in  4  rd.  ? 

13.  What  is  the  area  of  a  floor  12  ft.  by  ^  ft.  ? 

14.  Express  ^f-  in  simplest  form . 

15.  J  of  J  of  a  gallon  is  how  many  gills? 

16.  When  3  cents  will  buy  \  lb.  of  maple  sugar,  what  is  the 
cost  of  2  lb.  ? 

17.  |-  of  75  ft.  are  how  many  yards? 

18.  J  of  ^  of  a  yard  is  how  many  inches? 

19.  Frank  began  at  Chapter  XVII  in  his  book  this  morning 
and  has  read  12  chapters  to-day.  Express  in  Roman  notation 
the  number  of  the  last  chapter  which  he  read, 

20.  Multiply  2.04056  by  1000;  by  100 ;  by  10,000. 

21.  Divide  89,345  by  10 ;  by  1000 ;  by  10,000 ;  by  100. 

22.  42.86  =  4286^? 

23.  8903.4  =  8.9034  X? 

24.  3.984  x?  =  3984. 

246      _13  ^  246  x  13 
'    1000^100  ? 

26.  24.63  X. 029  =  2463  X  29 -^? 

27.  Name  and  describe  the  parties  to  an  account. 


REVIEW  AND  PRACTICE  131 

178.    Written 

1.  Express  in  Arabic  notation  eight  hundred  million  eighty 
thousand  eight,  and  seventy  thousandths. 

2.  Write  in  words  4040.0700. 

3.  Find  the  prime  factors  of  176;  482;  1260;  775;  385; 
1920. 

4.  Divide  the  product  of  36,  45,  20,  and  14  by  the  product 
of  80,  27,  35,  and  72. 

5.  Find  the   quotient  of  27  x  28  x  30  divided  by  18  x  35 
x36. 

6.  Find  the  G.  C.  D.  and  L.  C.  M.  of  182  and  196. 

7.  Find  the  smallest  number  that  will  exactly  contain  42, 
63,  and  105. 

8.  What  is  the  greatest  number   that  will  exactly  divide 
1176  and  1848  ? 

9.  Change  ^-^-^  to  simplest  form. 

10.  How  many  ninths  are  there  in  18  ? 

11.  Change  2^^  to  33ds. 

12.  8f-2lf  +  33-5,  =  ? 

13.  A  man  bought  5^  acres,  6^  acres,  and  10|  acres  of 
land.  He  then  gave  his  son  11|  acres.  How  many  acres  had 
he  left  ? 

14.  The  sum  of  two  fractions  is  ^^.  One  of  them  is  A-.  What 
is  the  other? 

15.  The  product  of  two  fractions  is  -^q.  One  of  them  is  ||. 
What  is  the  other  ? 

16.  Yesterday  you  bought  from  your  grocer  60  lb.  of  sugar 
at  5|^  per  pound,  60  clothespins  at  5/  per  dozen,  and  1  of  a 
barrel  of  flour  at  $4.64  per  barrel.     To-day  you  paid  the  bill. 


132  REVIEW  AND   PRACTICE 

and  the  grocer  receipted  it  and  gave  it  to  you  to  take  home. 
Write  a  copy  of  the  bill. 

17.  I  of  the  difference  between  -^j  and  -^^  is  what  ? 

18.  Find  the  cost,  at  $.Q2^  a  bushel,  of  the  seed  oats  for  a 
fieli  of  7^  acr^s,  if  the  farmer  sows  2|  bu.  on  an  acre. 

19.  Draw  a  plan  of  a  rectangular  lawn  4  rd.  by  2  rd.,  using 
-^Q  of  an  inch  for  a  foot.     That  is,  draw  it  to  the  scale  of  1'  to 

re  • 

20.  Find  the  cost  of  325|  bu.  of  wheat  at  il.l6|  per  bushel. 
(Express  i.l6|as  a  fraction  of  a  dollar.) 

21.  8 J  sq.  rd.  =  how  many  square  yards  ? 

22.  A  merchant  bought  |   of   a  piece  of   cloth  containing 
59^  yd.  at  37|^^  a  yard.     Find  the  cost. 

23.  How  long  will  it  take  a  motor  car  to  travel  87|  mi.,  if 
it  travels  12|  mi.  in  half  an  hour  ? 

24.  Divide  .1024  by  the  product  of  .064  and  .01. 

25.  How  many  cubic  yards  of  earth  must  be  removed  to  make 
an  excavation  123  ft.  by  24  ft.  by  12  ft.  ? 

26.  Make  a  problem  about  7  tons  of  coal,  $43.75,  and  28  tons 
of  coal.     Find  the  answer. 

27.  A  mile  is  5280  ft.     A  rod  is  16J  ft.     What  can  you  find 
from  these  numbers  ?     Find  it. 

28.  The  product  of  .025,  .85,  and  another  number  is  1.7. 
Find  the  other  number. 

29.  Find  the  number  of  cubic  inches  in  a  cake  of  ice  IJ  ft. 
by  2J  ft.  by  li  ft. 

30.  Find  the  number  of  cubic  feet  in  a  bin  6J  ft.  long,  4J  ft. 
wide,  and  3J  ft.  deep. 

31.  A  room  16 J  ft.  long  and  10|  ft.  wide  must  be  how  high 
to  contain  1425^  cubic  feet  of  air  ? 


PRODUCTS  AND  FACTORS  138 

PRODUCTS   AND   FACTORS  *  i^^ 

179.    Complete   each  of   the   following  statements   and   tell 
which  numbers  are  products  and  which  are  factors. 

1.  Since  1  rod  contains ft.,  4  rods  contain ft. 

2.  At  3J^  each,  8  cucumbers  will  cost cents. 

3.  At apiece,  f  47.50  will  buy  10  baseball  suits. 

4.  11.25  is  yV  of • 

5.  18  isf  of . 

6.  |of|  = . 

7.  .0Tof$25  = . 


8.  When  I  of   Paul's  earnings   for  a  week  are  $1.50,  he 
earns . 

9.  16.35-^.05= . 


10.  .75  of =  225bu. 

11.  .17|-  of  the  cost  of  a  piano  was  f  35.     The  piano  cost 


12.    t  of  a  farm  is  worth  f  1200.     The  whole  farm  is  worth 


"§" 


13.  A  city  lot  47  ft.  wide  cost  11950.50.     It  cost per 

front  foot. 

14.  54J  is times  2^-^. 

15.  13|^  yd.  of  broadcloth  at a  yard  cost  $51^. 

16.  A  rectangle  4  ft.  6  in.  by  8  ft.  3  in.  contains square 

inches. 

17.  7^V  = ^If. 

18.  19.25  will  pay  for  the  use  of  I when  $.05  will  pay 

for  the  use  of  one  dollar. 


134        STATEMENTS  AND  QUESTIONS  OF   RELATION 

180.        STATEMENTS  AND   QUESTIONS  OF  RELATION 

Queation. 
4x7J=? 
4  X  ?  =30 
?x7|  =  30 

|off  =  ? 
|of?  =  A 

Which  term  in  division  is  a  product  ? 

Which  terms  in  division  are  factors  ? 

When  the  factors  are  given,  what  must  be  done  to  find  the 
product  ? 

When  the  product  and  one  factor  are  given,  what  must  be 
done  to  find  the  other  factor  ? 

Every  problem  that  depends  on  the  relation  of  factors  and 
product  may  be  solved  by  one  of  these  operations ;  e.g. 

1.  a.  An  acre  of  land  costs  $80.  What  is  the  cost  of  12 
acres  ? 

Here  we  have  two  factors  given,  and  the  product  is  to  be 
found. 

Statement  of  Relation:  12  x  $80  =  cost  of  12  acres. 
12x$80  =  ? 
Solution:  12  x  $80  =  $960  Ans. 


Solution. 

4x7| 

=  30  Ans. 

30  H 

-4 

=n 

Ans. 

30 -i 

3 

=  4 

Ans. 

?x2 

_  9 

Ans. 

^ 

7 

14 

2 

3 

2 

9 

.5 

9 

x|  = 

.6 

Ans. 

14 " 

4 

7 
3 

3 

■7 

9 

6 

_  9 

T 

3 

Ans. 

X  -  = 

14  * 

7 

H 
2 

2 

^4 

STATEMENTS  AND  QUESTIONS    OF  RELATION        135 

h.    When  12  acres  of  land  cost  1 960,  what  is  the  cost  of  one 
acre  ? 

What  have  we  given  in  this  problem  ? 
What  is  to  be  found  ? 

Statement  of  Relation :  12  x  (cost  of  1  acre)  =  $960. 

12x?  =  !^960. 
.^  Solution :  ^  960  -^  12  =  $  80  A  ns. 

c.    If  1  acre  of  land  cost  i80,  how  many  acres  can  be  bought 
for  f  960? 

What  is  given  in  this  problem,  and  what  is  to  be  found  ? 
Statement  of  Relation  :  (the  number  of  acres)  x  $80  =  $960. 

?x  ISO  =  $960. 
Solution:  $960  -^  $80  =  12,  the  number  of  acres  Ans. 

We  may  make  use  of  the  relation  of  factors  and  product  in 
solving  problems  containing  fractions  ;  e.g, 

2.    a.    At  $80  an  acre,  what  is  the  cost  of  f  of  an  acre  of  land? 

Statement  of  Relation :  f  of  $  80  =  cost  of  |  of  an  acre. 

fof$80  =  ? 

10 

Solution  :  5  x  ^^  =  $50  Ans. 
^         1 

h.    When  |  of  an  acre  of  land  cost  $50,  what  is  the  cost  of 
one  acre  ? 

Statement  of  Relation:  f  x  (cost  of  1  acre)  =$50. 

fof?  =  $50. 

Solution:  $50^|  =  $80  Ans. 

c.    At  $80  per  acre,  what  part  of  an  acre  of  land  will  $50 
buy? 

Statement  of  Relation:  (number  of  acres)  x  $80  =  $50. 

?  of  $80  =  $50. 
Solution :  $50  -f-  $80  =  |g  =  f,  number  of  acres  Ans. 


136         STATEMENTS  AND  QUESTIONS  OF  RELATION 

3.    a.    What  is  the  volume  of  a  box  which  is  |  ft.  by  X  ft. 
by  2  ft.  ? 

Statement  of  Relation  :  ^  x  -^^  x  2'  =  volume. 

|xJjx2  =  ? 

3       7       2      7 
Solution  :   4  ^  To  ^  1  ~  8  ^^^*  ^^'^  ^'*^* 
2      4 

h.    What  must  be  the  length  of  a  box  |  ft.  wide  and  ^  ft. 
deep  to  contain  |  cu.  ft.  ? 

Statement  of  Relation :  (|  x  -5^)  x  length  =  J. 

|x^x?  =  f 

4 

Make  the  statements  for  finding  the  height  and  width  of  the 
box. 

In  each  of  the  following  problems  let  the  steps  be  taken  in 
this  order : 

•  First.  —  Read  the  problem  and  determine  what  is  given 
(two  factors,  or  product  and  one  factor),  and  what  is  to  be 
found  (the  product,  or  the  missing  factor). 

Second.  —  Make  the  statement  and  question  of  relation. 

Third.  —  Give  the  solution. 

181.    Written 

1.  99  is  how  many  times  12  ? 

2.  A  man  sold  his  farm  for  f  7200.     What  did  he  receive 
for  .125  of  the  farm  ? 

3.  36.48  is  how  many  times  .012  ? 


PRODUCTS  AND  FACTORS  137 

4.  How  many  feet  are  there  in  15  rd.  ? 

5.  Find  the  cost  of  98|  bu.  of  oats  at  40^  a  bushel. 

6.  How  much  money  have  I  if  |  of  it  is  ^100  ? 

7.  f  of  a  man's  salary  is  $1200.     What  is  his  salary  ? 

8.  16 J  is  the  product  of  33 J  and  what  other  number  ? 

9.  A  boy  spent  |  of  his  money  and  had  12.80  left.     How 
much  had  he  at  first  ? 

10.  A  man  owning  |  of  a  vessel  sold  |  of  his  share  for  14800. 
a.  What  part  of  the  vessel  did  he  sell?  h.  What  was  the 
whole  vessel  worth  ? 

11.  Wilfred  spent  $2.40,  which  was  |  of  his  money.  How 
much  money  had  he  ? 

12.  By  what  must  we  divide  5|  to  obtain  3|  ? 

13.  $75  will  pay  for  how  much  wheat  at  f  |  per  bushel  ? 

14.  The  multiplier  is  4^ ;  the  product  is  16^ ;  find  the 
multiplicand. 

15.  William  earns  $630  in  a  year,  which  is  ^  as  much  as 
his  father  earns.     What  can  you  find  ?     Find  it. 

16.  I  of  a  yard  of  cloth  cost  $|.  Make  the  question  and 
answer  it. 

17.  a.  A  farmer  bought  13|-  acres  of  land  at  $25|  per  acre. 
Find  the  cost.  h.  He  paid  for  it  in  wheat  at  $|^  a  bushel. 
How  many  bushels  of  wheat  were  required  ? 

18.  2.3  acres  of  land  for  $149.50  is  how  much  an  acre  ? 

19.  Find  the  price  of  a  yard  of  cloth  when  f  of  a  yard  cost 

$1.25. 

20.  a.  If  I  of  the  price  of  a  piece  of  timber  land  is  $4200, 
what  is  the  price  ?     h.  What  is  the  price  of  f  of  it  ? 


138  PRODUCTS  AND  FACTORS 

21.  a.  If  ^  of  a  ton  of  coal  'cost  $4,  what  is  the  price  per 
ton  ?     h.   How  much  coal  will  $151 J  buy  ? 

22.  A  man  owning  |^  of  a  store  sold  |  of  his  share  for  i  1000. 
a.  What  part  of  the  store  did  he  sell  ?  h.  What  was  the  whole 
store  worth  at  that  rate  ? 

23.  a.  12|  lb.  of  sugar  is  how  many  times  9|  lb.  ?  5.  If  9| 
lb.  cost  46|-  cents,  what  will  12|  lb.  cost  ?  c.  In  the  same  way, 
find  the  cost  of  16^  tons  of  coal,  when  2\  tons  cost  $12.60. 

24.  I  of  the  value  of  Mr.  Blank's  house  is  $2400.  a.  What 
is  the  house  worth  ?     h.  Find  |  of  its  value. 

25.  I  owned  f  of  a  farm  and  sold  ^  of  my  share,  a.  What 
part  of  the  farm  did  I  sell?  h.  If  I  received  $1248,  what  was 
the  farm  worth  ? 

26.  How  many  oranges  at  7J  cents  apiece  will  cost  as  much 
as  60  pears  at  2|^/  apiece  ? 

27.  .025  of  $5600  is  how  much  money  ? 

28.  .35  of  my  money  is  $700.     How  much  money  have  I  ? 

29.  A  grocer  buys  flour  at  $1.44  a  sack.  He  sells  it  so  as  to 
gain  .121  of  the  cost.     What  does  he  gain  on  a  sack  ? 

30.  A  merchant  sells  cloth  so  as  to  gain  $.20  on  a  yard. 
This  is  .40  of  the  cost.     Find  the  cost  of  a  yard. 

31.  Joseph  earned  $17,  and  used  .62|  of  it  to  help  his 
mother  pay  for  the  rent  of  her  house.  How  much  did  he  give 
toward  the  rent  ? 

32.  If  Mary  earned  $1.60  and  gave  her  mother  $.40,  what 
decimal  part  of  her  money  did  she  give  her  mother  ? 

33.  A  merchant  sold  cloth  for  cloaks  at  $8.30  per  yard. 
This  was  1.65  times  as  much  as  it  cost.     What  did  it  cost  ? 


PRODUCTS  AND  FACTORS  139 

34.  .22^2_  Qf  a  rod  is  how  many  feet  ? 

35.  Lewis  wants  a  bicycle.  In  21  days  he  can  earn  .42  of 
the  money  with  which  to  buy  it.  How  many  days  must  Lewis 
work  in  order  to  earn  the  bicycle  ? 

36.  Mr.  Johnson's  store  caught  fire,  and  his  goods  were 
damaged  by  smoke  and  water,  so  that  he  sold  them  for  ,35  of 
their  cost.  a.  What  did  he  receive  for  38  yd.  of  lace  that  cost 
$.25  a  yard?  h.  What  was  the  cost  of  a  coat  that  sold  for 
$  7  ?     c.  How  much  did  he  lose  on  a  table  that  cost  |15.00  ? 

37.  .49  of  a  number  is  19.6.  a.  Find  the  number.  5. 
Find  .62 J  of  the  number. 

38.  y\  of  a  certain  number  is  6^.  Find  .37  of  the  number. 
(How  many  statements  of  relation  ?) 

39.  Mr.  Byrne  bought  a  house  for  11200  and  sold  it  for 
$1800.     The  gain  was  what  decimal  part  of  the  cost  ? 

40.  3  inches  are  how  many  hundredths  of  3  feet  ? 

41.  I  borrowed  $250.  When  I  paid  it  back,  I  paid  my 
creditor  $250  and  .05  as  much  for  the  use  of  the  money.  How 
much  did  I  pay  in  all  ? 

42.  The  width  of  my  garden  is  48  feet.  The  width  is  .80  of 
the  length.     Find  the  length. 

43.  $|is  what  part  of  $10? 

44.  Eldred  sold  his  bicycle  for  $10.50,  which  was  .35  of  its 
cost.     What  did  it  cost  ? 

45.  There  were  50  words  in  the  spelling  lesson  and  Charlotte 
missed  two.  How  many  hundredths  of  the  words  did  she 
spell  correctly  ? 

46.  Raymond  shovels  the  snow  from  80  feet  of  sidewalk. 
After  shovelling  .25  of  the  walk,  how  many  feet  more  must  he 
shovel  ? 

47.  A  xMx?  =  4l 


140  PERCENTAGE 

PERCENTAGE 

182.  Decimals  in  hundredths  are  used  very  generally  in  busi- 
ness calculations.  The  merchant  calculates  his  gain  or  loss  as  a 
certain  number  of  hundredths  of  the  cost  of  the  goods.  Banks 
compute  interest  in  hundredths.  Agents  who  sell  goods  some- 
times figure  their  earnings  as  a  certain  number  of  hundredths 
of  the  selling  price  of  the  goods.  The  relations  of  numbers 
are  expressed  generally  in  hundredths.  • 

Per  cent  is  another  name  for  hundredths.  Six  per  cent  means 
six  hundredths ;  ten  and  one  half  per  cent  means  ten  and  one 
half  hundredths. 

Instead  of  writing  the  words  per  cent^  the  sign  %  is  used ; 
thus:  5%  of  $8  =  .05    x|8        =8.40. 

12i  %  of  4  in.  =  .12-1  x  4  in.     =  .50  in. 

4%  of  80  yd.    =    .04  x  80  yd.  =  3.20  yd. 

108%  of  $12  =  1.08x112      =iil2.96. 

Oral 

As  above,  tell  the  meaning  of  each  of  the  following  ex- 
pressions and  find  its  value: 

1.  8%  of  50  9.  1%  of  12100 

2.  50%  of  200  10.  5%  of  100  boys 

3.  12  %  of  100  miles  11.  80  %  of  20  horses 

4.  10%  of  60  sheep  12.  20%  of  400 

5.  50%  of  300  men  13.  33  J  %  of  900 

6.  2%  of  30  bushels  14.  10%  of  2000 

7.  25  %  of  64  days  15.  2|  %  of  30  days 

8.  10%  of  150  bushels  16.  7J%of$100 


PERCENTAGE  141 

17.  25%  of  200  minutes  20.    90%  of  1 60 

18.  33  J  %  of  175  21.    150%  of  20  pounds 

19.  5%  of  32  22.    300%  of  fl 

183.  The  statement  and  question  of  relation  aid  in  solving 
problems  in  percentage ;  e.g. 

1.  Donald  gave  his  mother  $3.75,  which  was  75%  of  his 
week's  wages.     How  much  a  week  did  he  receive  ? 

Read  the  question,  using  the  word  hundredths  in  place  of  pel 
cent. 

Statement  of  Relation:  75%  of  (Donald's  wages)  =  $3. 75. 

.75  of  ?  =  13.75. 

Solution:  $3.75  -e-  .75  =  $5  Am. 

The  statement  of  relation  shows  that  §3.75  is  a  product,  .75 
one  of  its  factors,  and  Donald's  wages  the  other  factor,  which 
we  are  to  find. 

2.  What  per  cent  of  1 38  is  $24.70  ? 

Reading  hundredths  instead  of  per  cent,  the  question  is, 
"How  many  hundredths  of  $38  is  $24.70?" 

Statement  of  Relation : of  $38  =  24.70. 

Solution :  $24.70  -r-  $38.  =  .65  or  65%  Ans. 

Observe  that  the  steps  in  solving  percentage  problems  are: 

a.  Read  the  question,  using  hundredths  in  place  of  per  cent. 

b.  See  which  terms  of  relation  are  given,  and  which  are  to 
be  found. 

c.  Give  the  statement  and  question  of  relation. 

d.  Make  the  solution. 

In  finding  the  per  cent,  or  number  of  hundredths,  how  many 
decimal  places  must  there  be  in  the  quotient?  Then  how 
must  the  number  of  decimal  places  in  the  dividend' compare 
with  the  number  of  decimal  places  in  the  divisor  ? 


142  PERCENTAGE 

Therefore, 

Before  dividing^  to  find  per  cents,  arrange  the  dividend  and 
divisor  so  that  the  dividend  contains  two  more  decimal  places  than 
the  divisor.  This  may  he  done  hy  annexing  ciphers  to  one  or  the 
other  of  these  terms,  as  may  he  necessary. 

If  the  quotient  is  not  exact  when  two  decimal  places  have  been 
reached,  express  the  remainder  as  a  common  fraction,  in  the  quo- 
tient; thus  : 

3.  1.65  T.  is  what  per  cent  of  4.2  T.  ? 

Statement  of  Relation : of  4.2  T.  =  1.65  T. 

Solution :  1.65  T.  -^  4.2  T.  =  .39f  or  39f  %. 

.8911  =  .39f  or  39^  %  Ans. 

4.2)1.6-50 

126 

390 

3  78 

12 

4.  Mr.  Moore  earns  $140  a  month  and  saves  $84.  His 
earnings  are  what  per  cent  of  his  savings  ? 

Statement  of  Relation:  of  $84  =  $140. 

Solution ;  fiUO  -^  $84  =  1.66f  or  166|%  Ans. 

____L66|f  =  1.66|  or  166f  %  Ans, 
$84.)$  140.00 
84 
560 
504 
560 
504 
56 

To  THE  Teacher.  —  It  is  well  to  follow  this  plan  until  pupils  acquire  a 
considerable  degree  of  skill  in  the  process.  After  pupils  have  become 
well  grounded  in  the  fundamental  idea  that  per  cent  and  hundredths  are  iden- 
tical as  expressions  of  ratio,  it  may  be  desirable  to  give  practice  in  expressing 
fractional  parts  of  1%  as  decimal  approximations,  as  8.33%  instead  of  8^%. 


PERCENTAGE  143 

Written 

Solve  the  following,  using  the  steps  indicated  above  : 

5.  Find: 

a.  40  %  of  120  h.  35  %  of  700  pupils 

b.  121%  of  160  lb.  I  1%  of  190 

c.  18|%  of  365  y.  16|  %  of  66  miles 
c?.  36%  of  250  yd.  k.  48%  of  $8.50 

e.    81%  of  360  L    100%  of  $24.70 

/.    6|%of64bu.  m,   200%  of  118.79 

g.    37%  of  $16.50 

6.  a.    6.8  is  17%  of  what? 

b.  $3.95  is  5%  of  what? 

c.  $289  is  50%  of  what? 

d.  75  doz.  is  2^  %  of  how  many  dozen  ? 

e.  5  is  5  %  of  what  ? 

7.  a.   What  per  cent  of  $240  is  $80  ? 

b.  150  is  what  per  cent  of  900  ? 

c.  What  per  cent  of  $113  is  $39.55  ? 

d.  Find  what  per  cent  495  years  is  of  825  years. 
€.  Find  what  per  cent  12.96  feet  is  of  96  feet. 

8.  45  bushels  are  90  %  of  what  ? 

9.  3 J  miles  is  70  %  of  what  distance  ? 

10.  What  is  81%  of  690  1b.? 

11.  What  per  cent  of  $  920  is  $230  ? 

12.  15  minutes  are  what  per  cent  of  25  minutes^ 

13.  a.    1  quart  is  what  per  cent  of  4  quarts  ? 
b.    1  quart  is  what  per  cent  of  1  gallon  ? 

14.  Find  181%  of  362. 

15.  Of  what  number  is  3.71  twenty-five  per  cent? 


144  PERCENTAGE 

16.  15  9J  of  a  number  is  10.50.     What  is  the  number  ? 

17.  Whatisl|%  of  640? 

18.  37  J  %  of  what  equals  15  lb.  ? 

19.  What  per  cent  of  85  lb.  is  17  lb.  ? 

20.  35  yd.  are  what  per  cent  of  105  yd.  ? 

21.  Henry  raised  80  chickens  and  sold  75  %  of  them.  How 
many  chickens  did  he  sell? 

22.  Mary  has  an  allowance  of  f  25  a  year.  If  she  puts  12% 
of  it  in  the  bank  each  year,  how  much  will  she  put  in  the 
bank  in  5  years  ? 

23.  There  were  50  words  in  the  spelling  test.  Dorothy 
spelled  98  ^^  of  them  correctly.  How  many  did  she  have 
right  ? 

24.  In  a  school  of  660  pupils,  55  %  were  girls.  How  many 
were  girls? 

25.  45  %  of  a  class  of  40  pupils  were  boys.  How  many 
were  girls  ? 

26.  60  %  of  a  class  were  boys.  a.  If  there  were  21  boys, 
how  many  pupils  were  there  in  the  class?  h.  How  many  were 
girls  ? 

27.  In  a  class  of  50  pupils,  there  were  27  girls.  What  per 
cent  of  the  class  were  boys  ? 

28.  In  one  day  a  grocer  sold  14|  %  of  a  barrel  of  sugar, 
containing  350  lb.     How  many  pounds  did  he  sell  ? 

29.  Mr.  Williams  has  40  acres  of  timber  land.  This  is  25  % 
of  all  his  land.     How  many  acres  has  he  ? 

30.  .375  is  what  per  cent  of  .875  ? 

31.  A  man's  house  and  furniture  are  insured  for  $3600. 
40  %  of  this  insurance  is  on  the  furniture.  What  is  the  insur- 
ance on  the  furniture  ? 


PERCENTAGE  145 

32.  Mr.  French  borrowed  $  725  of  Mr.  Rich,  and  paid  him 
6  %  of  that  sum  for  the  use  of  it.  How  much  was  paid  for 
the  use  of  the  1 725  ? 

33.  A  man  borrowed  some  money  and  paid  1 75  for  the  use 
of  it.  If  that  was  5  %  of  the  sum  borrowed,  how  much  was 
borrowed  ? 

34.  A  bushel  of  potatoes  weighs  60  lb.  If  75  %  of  this  is  water, 
how  many  pounds  of  water  are  there  in  a  bushel  of  potatoes  ? 

35.  It  costs  a  man  $1200  to  support  his  family  for  a  year.  If 
this  is  80  %  of  his  salary,  what  is  his  salary  ? 

36.  920  lb.  or  23  %  of  a  load  of  grain  is  wheat.  How  many 
pounds  does  the  load  weigh  ? 

37.  a.  In  a  baseball  game,  Fred's  team  scored  14  runs. 
If  Fred  made  2  runs,  what  per  cent  of  the  14  runs  did  he 
make  ?  5.  If  the  other  team  made  6  runs,  what  per  cent  of  all 
the  runs  did  Fred's  team  make  ? 

38.  A  bat  and  ball  cost  $1.50.  If  33  J  %  of  this  sum  was 
paid  for  the  bat,  what  did  the  ball  cost  ? 

39.  A  huckster  bought  berries  at  8^  a  quart  and  sold  them 
at  12^.     His  gain  was  what  per  cent  of  the  cost  ? 

40.  35  gallons  of  Jersey  milk  contained  8.05  gallons  of 
cream.     What  per  cent  of  the  35  gallons  was  cream  ? 

41.  A  man  lost  $127.50  by  selling  a  village  lot.  If  this 
was  15  %  of  the  cost,  what  did  the  lot  cost  ? 

42.  A  merchant  sold  goods  for  $25.63  less  than  the  marked 
price.  If  this  reduction  was  10  %  of  the  marked  prioe,  what 
was  the  marked  price  ? 

43.  A  grocer  bought  a  crate  of  cherries  containing  32  quarts 
for  $2.88.  He  sold  them  at  12  cents  a  quart.  His  gain  was 
what  per  cent  of  the  cost  ? 


146 


REVIEW   AND  PRACTICE 


REVIEW   AND  PRACTICE 


60' 

Popcorn                                                 ^ 

Sweet  Com                                             * 

Tomatoes                                                "^ 

Cucumbers 

I                         Squash                    *^ 

;                i 

Early     •   „    ,.  ^      1        Late 
,  _         1  Radishes  \     , 
Lettuce   j                   j    Lettuce 

1                           Spinach                        j  Peppers"^ 

Cabbages                                                 « 

Cauliflower                                               w 

ti             i 

t\       Parsnips       i       Salsify 

Carrots    \       Onions       \                Beets               « 

i          1             ! 

Flowers                                                « 

This  is  the  phin  of  a  real  garden  which  Joseph  and  his  father 
cultivated.     All  the  st^iteraents  in  the  exercises  are  facts. 

Oral 

1.  Take  your  rule  and  find  out  the  scale  of  the  plan. 

2.  Find   the   number   of   square   feet   of    land    planted    to 
radishes  ;  to  late  lettuce  ;  to  spinach;  to  onions  ;  to  beets. 


REVIEW  AND  PRACTICE 


147 


n 


184.    Written 

1.  The  corn  was  planted  so  that  each  hill  occupied  one  square 
yard  of  ground. 

a.  How  many  hills  of  pop  corn  were  there  ? 

b.  How  many  hills  of  sweet  corn  ? 

2.  There  were  six  hills  of  cucumbers,  and  the  same  number 
of  hills  of  squash.  How  many  square  feet  of  land  did  each  hill 
occupy  ? 

3.  There  were  15  tomato  plants.  How  much  space  did  each 
plant  have  ? 

4.  The  cabbage  and  cauli 
flower    plants    each    had 
square  feet  of   space.     How 
many  plants  were  there  ? 

5.  a.  What  per  cent  of 
the  entire  garden  is  planted 
to  flowers  ?  5.  To  cabbage 
and  cauliflower  ?  e.  To 
spinach  ? 

6.  The  family  table  was 
supplied  with  vegetables 
from   the   garden,   a   correct 

account  of  them  being  kept.     Those  not  needed  at  home  were 
sold  by  Joseph  during  his  spare  time,  mornings  and  Saturdays. 

Find  the  value  of  each  of  the  following  productions : 

a.  Sweet  corn     ....     312  ears  $.13  per  dozen. 

b.  Tomatoes       ....     6  bu.  $.04  per  quart. 

c.  Popcorn 54  1b.  $.041  per  lb. 

d.  Cucumbers    ....     219  3  for  10  cents. 

e.  Squash 88  1b.  3^^  per  lb. 


*.:•  :;*i^ 


148  REVIEW  AND  PRACTICE 

/.  Cauliflower    ....  20  heads  $.18  apiece. 

g.  Lettuce  (early)       .     .  100  heads  $.04  apiece. 

h.  Lettuce  (late)     ...  70  heads  2  for  5  cents. 

i.  Spinach 4  bu.  18^  per  pk. 

j.  Radishes QQ  bunches  3  for  5  cents. 

k.  Parsnips 1  bu.  25/  per  pk. 

I.  Salsify 25  bunches  5/. 

m.  Peppers Ill  16/  per  doz. 

n.  Young  onions     ...  40  bunches  2  for  5  cents. 

0.  Ripe  onions    ....  J  bu.  4/  per  qt. 

p.  Cabbage    .....  22  heads  9/  apiece. 

q.  Carrots 22  bunches  2  for  5  cents. 

r.  Beets 30  bunches  2  for  5  cents. 

«.  Beets  (full  grown)       .  3  pk.  60/  per  bu. 

7.  What  was  the  total  value  of  these  productions? 
Make  other  problems  using  the  numbers  given  above. 

8.  The  first  ripe  tomatoes  were  gathered  on  the  14th  day 
of  August  and  the  latest  on  the  17th  day  of  October.  How 
many  days  did  they  last  ? 

9.  A  crop  of  peas  was  grown  before  the  cabbage  and  cauli- 
flower plants  were  set  out.  Peas  were  raised,  also,  on  either  side 
of  the  rows  of  tomatoes,  cucumbers,  and  squash  before  those 
vines  began  to  spread.  The  crop  of  peas  was  2|^  bu.  and  they 
were  worth  33/  a  peck.  Find  the  value  of  the  crop  of  peas 
raised  here. 

10.  Beans  were  planted  between  the  rows  of  corn  and 
matured  before  the  corn  was  high  enough  to  shut  out  the  sun. 
The  three  bushels  of  string  beans  thus  raised  were  worth  how 
much  at  5/  a  quart? 

II.  Some  pumpkin  seeds  were  planted  with  the  seed  corn,  and 
yielded  27  pumpkins  worth  12  cents  apiece.  What  were  they 
all  worth? 


REVIEW  AND  PRACTICE  149 

12.  Late  sweet  corn  was  planted  where  the  radishes,  spinach, 
lettuce,  and  part  of  the  peas  had  grown,  and  yielded  148  ears 
worth  12  cents  a  dozen.     What  was  the  crop  worth? 

13.  Adjoining  this  garden  is  a  space  of  the  same  size  de- 
voted to  fruit,  flowers,  and  perennial  vegetables.  The  flowers 
were  used  for  the  adornment  of  the  home  and  the  pleasure  of 
giving  them  to  others.  The  extra  fruit  and  vegetables  were 
sold. 

Find  the  total  value  of  the  following: 

Asparagus,  averaging  one  bunch  per  day  from  May  1  to  July  1 
at  9  cents  a  bunch. 

Twenty  bundles  of  pie-plant  at  5  cents  per  bundle. 
Three  pounds  of  sage  at  35  ^  per  pound. 
Fifty-three  baskets  of  grapes  at  20/  per  basket. 
Six  and  one  half  bushels  of  plums  at  20  /  per  peck. 
Four  bushels  of  cherries  at  12/  per  quart. 
Two  bushels  of  pears  at  30/  per  peck. 
Two  and  one  half  bushels  of  peaches  at  8/  per  quart. 
Two  bushels  of  currants  at  9/  per  quart. 

14.  The  expenditures  were  : 

For  fertilizers 16.50 

For  seeds 1.38 

For  Bordeaux  mixture  for  spraying  trees,  etc.     ...  .40 

For  Paris  green  for  spraying .30 

For  trees,  vines,  and  plants 2.38 

For  implements  worn  out .65 

Joseph  received  as  his  share  $25.20,  which  was  25%  of  the 
,value  of  all  that  was  raised,  after  the  expenses  were  paid. 
a.    What  was  the  value  of  the  produce  less  expenses? 
5.    What  was  the  total  value  of  the  produce? 


150     PER  CENTS  EQUIVALENT   TO   COMMON  FRACTIONS 

PER   CENTS   EQUIVALENT   TO   COMMON   FRACTIONS 

185.  All  percentage  problems  involving  the  relation  of  prod- 
uct and  factors  may  Jbe  solved  in  decimals.  But  in  many  cases, 
as  we  shall  see,  the  work  may  be  shortened  by  changing  the  per 
cents  to  common  fractions. 

Oral 

1.  The  whole  of  anything  is  how  many  hundredths  of  it  ? 
What  per  cent  of  it  ? 

2.  J  of  anything  is  how  many  hundredths  of  it?    ^?    -|? 

tV?  I?  I?  *?  V  V  V  f?  V  V  F  V  V  iV? 

3.  What  common  fraction  is  the  same  as  .10?  .20?  .30? 
.40?  .50?  .60?  .70?  .80?  .90?  .25?  .33J?  .14f? 
.62 J?  .371?  .66|?  .121?  .87^?  .75?  .16f?  .83^?  .08^? 
.05? 

4.  What  per  cent  is  the  same  as  ^?  ^?  J?  |?  J?  |?  \?  ^V- 
A?  2V?  ^?  2V?  F  I?  V  V  i?  f?  V  F 

5.  Learn  this  table : 


1  =  50%  i  =  BSl%  |  =  62|% 

1  =  25%  i  =  ^H%  1  =  87^% 

1=76%  J  =  16|%  ^=10% 

i  =  20%  f  =  83i%  ^  =  8^% 

1  =  40%  |  =  14|%  5'(r=5% 

1  =  60%  i  =  12i%  TV  =  6i% 

1=80%  1=371%  5>j  =  4% 


PERCENTAGE 


151 


Answer  the  following  questions,  using  common  fractions  instead 
of  decimals  when  it  is  easier  to  do  so. 

6.    Find: 


k.  6  J  %  of  16  days 

I.  60  %  of  50  ft. 

m.  371%  of  64 

n.  87|  %  of  96 

0.  83  J  %  of  18  trees 

p.  121%  of  600 

q,  331%  of  60  years 

r.  4%  of  150 

«.  81  %  of  3600  people 

t.  25  %  of  836  miles 


a.    331%  of  12 
5.    10%  of  200 
c.    16f%  of  30  da. 
d. .  81  %  of  144  sq.  in. 
e,    75  %  of  28  gal. 
/.    66|%of  27cu.  ft. 
g,    20%  off 
h.    40%  of  1100 
z.    90%  of  $200 
y.    62 J  %  of  24)2^ 

7.  9  is  25  %  of  what  number  ? 

8.  32  is  50  %  of  what  number  ? 

9.  75  is  10  %  of  what  number  ? 

10.  4  is  what  per  cent  of  16  ?     (^  =  what  per  cent  ?) 

11.  6  is  what  per  cent  of  9  ?     (f  =  |  ==  what  per  cent  ?) 

12.  1  is  what  per  cent  of  16  ? 

13.  3  months  are  what  per  cent  of  18  months  ? 

14.  3  books  are  what  per  cent  of  15  books  ? 

15.  17  are  what  per  cent  of  1 84  ? 

16.  9  men  are  75  %  of  how  many  men  ? 

17.  10  min.  are  16|%  of  what? 

18.  2  is  what  per  cent  of  40  ? 

19.  80%  of  35^  is  what? 

20.  20  %  of  my  money  is  16  ^.     How  much  money  have  I  ? 

21.  8  cents  are  66|  %  of  what  ? 


152  PERCENTAGE 

22.  Frank  is  15  years  old.  Julia's  age  is  16|  %  of  Frank's. 
How  old  is  Julia? 

23.  Will  is  8  years  old  and  Ethel  is  7.  Elhel's  age  is  what 
per  cent  of  Will's? 

24.  I  gained  f  7  in  selling  my  watch.  If  that  was  12  J  %  of 
the  cost,  what  did  it  cost  ? 

25.  Out  of  25  words,  Charlie  missed  one.  What  per  cent 
of  the  words  did  Charlie  miss  ? 

26.  Edith  had  9  examples  right.  If  there  were  10  in  the 
lesson,  what  per  cent  of  them  did  she  have  right  ? 

27. — ■  =  how  many  hundredths  ? 

100       100  ^ 

28.  100  %  —  90  %  =  what  per  cent  ? 

29.  If  I  spend  90  %  of  my  money,  what  per  cent  do  I  have 
left  ? 

30.  Having  spent  90  %  of  my  money,  I  had  %  2  left.  How 
much  money  had  I  at  first  ? 

31.  A  grocer  sold  90  %  of  a  barrel  of  sugar  and  had  35  lb. 
left.     How  many  pounds  did  the  barrel  contain  at  first  ? 

32.  Hubert  gave  away  75  %  of  his  apples  and  had  6  left. 
How  many  had  he  at  first  ? 

33.  Harry  took  a  silver  dollar  to  the  store  and  bought  2  lb. 
of  cheese  at  18  ^  a  pound.  On  the  way  home  he  lost  12J  % 
of  the  change.     How  much  change  did  he  lose  ? 

34.  A  boy  sold  66f  %  of  his  chickens  and  kept  20.  What 
per  cent  of  them  did  he  keep  ?  How  many  had  he  at  first  ? 
How  many  did  he  sell  ? 

36.    Sarah  answered    correctly  80  %   of   the  questions  that, 
came  to  her  and  missed  one.     How  many  questions  came  to 
her  ?     How  many  did  she  answer  correctly  ? 


PERCENTAGE  153 

36.  Alfred  attended  school  98  %  of  the  days  of  the  term. 
If  he  was  absent  2  days,  how  many  days  were  there  in  the  term  ?. 

37.  A  boy  was  sick  and  stayed  out  of  school  5  days  in  one 
month.  If  there  were  20  days  of  school  in  that  month  what 
per  cent  of  the  time  did  he  attend  school  ? 

38.  62 1  %  of  a  cask  of  vinegar  leaked  out.  If  there  were  15 
gallons  left,  how  many  gallons  did  the  cask  hold  ? 

39.  .20  +  .10  =  how  many  hundredths? 

40.  20  %  +  10  %  =  how  many  per  cent  ? 

41.  A  butcher  bought  200  lb.  of  beef.  He  sold  20  %  of  it 
to  one  man  and  10  %  of  it  to  another.  What  per  cent  of  the 
beef  did  he  sell  ?  What  per  cent  was  left  ?  How  many  pounds 
were  left  ? 

42.  What  per  cent  of  this  oblong  is  -4.? 
J5?  O?  m  If  A  is  50  sq.  in.,  what  is 
Bl  (7?  i>?  El  What  per  cent  of  the 
oblong  are  A^  jB,  and  Q  together  ? 

43.  In  a  shipwreck,  \  of  the  crew  were 

lost.     What  per  cent  were  saved  ?     If  80  men  were  saved,  how 
many  were  lost  ? 

44.  Alice,  having  read  75  %  of  a  book,  has  50  pages  yet  to 
read.     How  many  pages  does  the  book  contain  ? 

45.  A  man  has  traveled  60  %  of  the  distance  from  New  York 
to  Chicago  and  has  400  miles  yet  to  travel.  What  is  the  whole 
distance  ? 

46.  When  25  days  of  the  month  of  November  are  past,  what 
per  cent  of  the  month  is  yet  to  come  ? 

47.  If  your  schoolroom  is  40  feet  long  and  30  feet  wide,  its 
width  is  what  per  cent  of  its  length  ?  Its  length  is  what  per 
cent  of  its  width  ? 


A 

B 

C 

D 

E 

154  PERCEKTAGE 

186.     Written 

1.  An  army  of  19,000  men  went  to  the  front.  12|  %  of 
them  were  killed  in  battle,  and  25  %  of  them  died  of  wounds 
and  sickness.     How  many  were  left  ? 

2.  A  collector  for  a  newspaper  started  out  with  bills 
amounting  to  ^840.  He  collected  $156.  a.  What  per  cent 
of  the  bills  did  he  collect  ?  b.  What  per  cent  did  he  fail  to 
collect  ? 

3.  35  %  of  the  apples  in  an  orchard  were  unfit  for  market 
and  could  not  be  sold.  If  1300  bushels  were  sold,  what  was 
the  entire  yield  ? 

4.  39  %  of  the  4700  blossoms  on  a  cherry  tree  were  blasted 
and  the  rest  became  fruit.  How  many  cherries  did  the  tree 
bear  ? 

5.  98  %  of  the  men  in  a  certain  city  can  read  and  write. 
If  there  are  1398  men  who  cannot  read  and  write,  how  many 
men  are  there  in  the  city  ? 

6.  23  %  of  the  men  in  a  certain  city  work  in  factories.  If 
there  are  9200  men  who  work  in  factories,  how  many  men  are 
there  in  the  city  ? 

7.  45  %  of  a  jeweler's  goods  were  stolen,  a.  If  he  had 
i  16,500  worth  of  goods  left,  what  was  the  entire  stock  of  goods 
worth  ?     h.  How  many  dollars'  worth  were  stolen  ? 

8.  In  an  orchard  of  3600  trees,  25  %  were  pear  trees,  15  % 
peach  trees,  10  %  plum  trees,  and  the  rest  apple  trees.  How 
many  apple  trees  were  there  ? 

9.  My  gas  bill  for  one  month  was  f  1.80.  Five  per  cent  of  it 
was  deducted  for  prompt  payment.  What  was  saved  by  pay- 
ing promptly  ? 

10.    A  merchant  bought  a  piece  of  cloth  for  f  65  and  sold  it 
for  130  %  of  its  cost.     What  did  he  receive  for  it  ? 


PERCENTAGE 


155 


11.  This  load  of  hay  weighs  2200  lb.,  the  wagon  1200  lb., 
and  the  team  2600  lb.  a.  The  weight  of  the  hay  is  what  per 
cent  of  the  entire  weight  ?  h.  The  weight  of  the  wagon  is 
what  per  cent  of  the  entire  weight?  c.  The  weight  of  the 
team  is  what  per  cent  of  the  entire  weight  ?  d.  How  many 
tons  must  the  bridge  support  ? 

12.  a.  In  1905  the  Chicago  baseball  team  won  92  games 
and  lost  60.  What  per  cent  of  all  the  games  played  did  they 
win  ?  h.  The  Boston  team  won  78  and  lost  74.  What  per 
cent  of  all  the  games  did  they  lose  ?  c.  The  New  York  team 
played  149  games  and  won  47^9^"^  %  of  them.  How  many  games 
did  they  lose  ? 

13.  By  selling  paper  at  150  %  of  its  cost,  a  stationer  receives 
90  cents  a  package  for  it.  What  is  the  cost  of  a  package  of 
this  paper  ? 

14.  Twelve  pounds  of  seed  for  a  lawn  contained  21  lb.  of 
white  clover  seed.  What  per  cent  of  the  mixture  was  white 
clover  seed  ? 


156  A  FRACTION  IN  THE  MULTIPLICAND 

15.  a.  If  a  pine  plank  weighs  45  lb.  and  an  oak  plank  of  the 
same  size  weighs  72  lb.,  the  weight  of  the  pine  is  what  per  cent 
of  the  weight  of  the  oak?  h.  The  weight  of  the  oak  wood  is 
what  per  cent  of  the  weight  of  the  pine  ? 

16.  A  grocer  bought  100  lb.  of  soda  for  $3.50  and  sold  it 
for  142|^  %  of  its  cost.     What  did  he  receive  a  pound  for  it  ? 

A  FRACTION  IN  THE  MULTIPLICAND 

187.  In  multiplying  a  large  mixed  number  by  an  integer, 
time  may  often  be  saved  by  multiplying  the  whole  number  and 
the  fraction  separately,  then  adding  the  products,  thus : 

314|  X  5  =  ? 

3141 
5 

3|  =  I  X  5 
1570    =  314  X  5 
1573f  =  3141  X  5 

Multiply: 

1.  248f  by  5  6.  m^-^  by  8  li.  224J^  by  9 

2.  39jl-  by  3  7.  35f  by  7  12.  42^^  by  12 

3.  42^  by  6  8.  49^^  by  13  13.  65|-|  by  16 

4.  8501  by  15  9.  207fby3  14.  201ff  by  16 

5.  292  by  11  10.  38^gbyl6  15.  431|J  by  72 

188.  In  division,  if  either  dividend  or  divisor  contains  a 
common  fraction  that  cannot  be  easily  reduced  to  a  decimal,  it 
is  sometimes  helpful  to  multiply  both  dividend  and  divisor  by 
the  denominator  of  the  fraction,  thus  making  both  dividend 
and  divisor  integers,  or  simple  decimals ;  e.g. : 


A  FRACTION  IN   THE  DIVISOR  167 


.05f    148.74 

Multiplying  both  dividend  and  divisor  by  7, 

2814. 

Quotient 

.37    1041.18* 

(Multiplying   the  dividend   and 

74 

divisor   by  the   same   number 

301 

affects  the  quotient  how  ?) 

296 

51 

37 

148 

148 

>. 

1.    2.295  -^  .05| 

6. 

.3125  -*-  .02^ 

2.    10.44 -J- .04| 

7. 

787.2  ^  .931- 

3.   36  -^  .13J 

8. 

1.024  -^  .005 J 

4.    96.9  ^  .15| 

9. 

218.24  -^  1.13| 

5.    .0256-1- .071 

10. 

385.35  -h  5.2f 

11.    Find  the  number,  of  which : 

a.    72  is  5{  % 

/• 

701.4  is  4|  % 

h.    10.5  is  11^% 

9- 

284.4  is  1051  % 

c.    24.64  is  391% 

h. 

5.775  is  116f  % 

d,   12.834  is  14f% 

i. 

.3155  is  901  cj^ 

e.    1263  is  171% 

h 

.833  is  108y\  % 

12.    Mr.  Fitch  rained  14.60  in  se 

illinof  a  waffon.     This  was 

6|  %  of  its  cost.    What  was  the  cost  of  the  wagon  ? 

13.  The  average  attendance  in  a  certain  school  was  640 
pupils.  If  this  was  91  f  %  of  the  number  registered  in  the 
school,  how  many  were  registered  ? 


158  REVIEW  AND  PRACTICE 

REVIEW  AND  PRACTICE 
189.    Oral 

1.  What  number  is  composed  of  5  units,  7  tens,  and  3 
thousands  ? 

2.  Read  XLIV;  CCLXII;  DCXCI;  MC  IVIII;  CDLIV. 

3.  G-ive  results  rapidly^  adding  or  subtracting  the  tens*  figures 
first:  36  +  45;  29  +  32;  57  +  76;  93+28;  93  -27;  84-45; 
72  +  39. 

4.  Grive  quickly  the  number  of: 

a.  Quarts  in  98  pt.  j.  Feet  in  2  rd. 

b.  Pecks  in  28  bu.  k.  Dollars  in  36,000  cents. 

c.  Hours  in  a  week.  I.  Gills  in  a  gallon. 

d.  Seconds  in  1  hour.  m.  Days  in  two  common  years. 

e.  Inches  in  2  yd.  n.  Tons  in  1600  lb. 

/.    Square  inches  in  2  sq.  ft.     o.    Square  rods  in  10  A. 
g.    Square  yards  in  450  sq.  ft.   p.    Yards  in  10  rd. 
h.    Cubic  feet  in  2  cu.  yd.  q.    Days  in  14  wk. 

i.   Dimes  in  |15.  r.    Days  in  a  summer. 

«.    Cubic  inches  in  a  box  5  in.  by  2  in.  by  1  in. 

5.  A  half  dollar,  a  quarter,  2  dimes,  and  a  nickel  are  how 
many  cents  ? 

6.  i  +  i  +  A  =  ?  10.  15-J  =  ? 

7.  3x8x?  =  48  *  11.  18- If  =  ? 

8.  8x9  =  6  X?  12.  5|  +  13f=? 

9.  88h-?  =  8  13.  7|-f=:? 

14.  When  36  men  can  earn  a  sum  of  money  in  15  da.,  how 
long  will  it  take  12  men  at  the  same  wages  to  earn  the  same 
amount  ?    9  men  ?     6  men  ?     72  men  ? 


REVIEW  AND  PRACTICE  159 

15.  If  6  men  earn  8  dollars  in  a  certain  time,  how  many  men 
can   earn  $  16  in  the  same  time  at  the    same  wages  ?     $  32  ? 

164?    14? 

16.  If  10  men  earn  $  200  in  8  days,  how  many  dollars  will 
twice  as  many  men  earn  in  that  time  at  the  same  rate  ?    ' 

17.  fof8bu.=  ?  21.  36  is  y9^  of  what? 

18.  I  of?  =  14.  22.  iof^f  =  ? 

19.  27  is  -fj  of  what  ?  23.  f  of  what=  15  qt.  ? 

20.  What  part  of  18  is  15?  24.  |  of  what  =  ^y? 

25.  What  are  the  prime  factors  of  84? 

26.  Name  two  numbers  that  are  prime  to  12. 

27.  How  many  42ds  are  there  in  |? 

28.  What  is  the  least  number  that  exactly  contains  6, 15,  and 
20? 

29.  Name  three  numbers  of  which  7  is  an  exact  divisor. 

30.  Give  two  composite  numbers  that  are  prime  to  each  other. 

31.  How  may  we  tell  whether  a  number  is  prime  or  not  ? 

32.  What  is  the  greatest  number  that  will  exactly  divide  26 
and  39  ? 

33.  What  is  the  smallest  number  that  6,  8,  12,  and  16  will 
divide  ? 

34.  Crive  results  at  sight : 

a.  362  --  10  /.  14  x  200  k.  .06  of  500 

h.  4900-1000    ^.  99-f-.l  I.  lof =  7| 

e.  29  X  .01  h.  .224  T  = lb.  m.  5%  of  25 

cZ.  834-10,000     z.  23,400-200  n.  121%  of =  7 

e,  29x1000         y.   .12x50  o.  ^V  = % 

35.  What  per  cent  of  a  ton  is  400  lb.  ? 

36.  Compare  .84  x  25  with  84  x  .25. 


160  REVIEW  AND  PRACTICE 

37.  (16  +  4)  X  (43 -23)  =  ? 

38.  What  must  we  do  with  |,  |,  and  |  to  find  out  which  is 
greatest  ? 

39.  Numerate  23516.00562. 

40.  Compare  the  values  of  |,  |^|,  and  i^. 

41.  Find  the  cost  of: 

a.    64  lb.  of  pork  at  12 J  ^ 

h,    600  boxes  of  berries  at  8|  ^ 

c.  96  gal.  of  molasses  at  50^ 

d.  16  doz.  oranges  at  37 J  ^ 

e.  54  yd.  of  matting  at  33 J  ^ 
/.    48  1b.  butter  at  25^ 

g.    16  knives  at  62 J  ^ 

h.   2  doz.  sleds  at  87|^  apiece 

^.    80  lb.  rice  at  6^^ 

42.  How  many  bushels  of  beets  will  §12  buy  at  33^^  a 
bushel? 

43.  f  16  will  rent  a  boat  for  how  many  hours  at  16^  an  hour  ? 

44.  What  change  will  be  left  from  a  f  20  bill  after  paying  for 
12  hours'  labor  at  37^  ^  an  hour  ? 

45.  At  16|^  a  dozen,  how  many  ears  of  corn  will  $2  buy  ? 

46.  ^  of  ^  lb.  = oz. 

47.  What  is  the  area  of  a  rectangle  5^  inches  by  4  inches  ? 

48.  What  is  the  cost  of  a  dozen  tomato  plants  at  the  rate  of 
4  for  5  cents? 

49.  At  10^  a  dozen,  how  many  sheets  of  sandpaper  will  5^ 
buy? 


REVIEW  AND  PRACTICE  161 

50.  A  man  spent  20  %  of  his  salary  for  rent,  10  %  for  cloth- 
ing, 5  %  for  fuel  and  light,  25  %  for  food,  and  20  %  for  other 
things.  What  per  cent  of  his  money  did  he  spend  ?  What  per 
cent  did  he  save  ?  If  he  saved  |400  a  year,  what  was  his 
salary  ? 

51.  What  is  120%  of  5  miles ? 

190.    Written 

1.  Write  in  figures  forty-two  thousand,  and  two  hundred 
five  ten-thousandths. 

2.  209  X  87,000  0,6%^). 

3.  235,404  -^  468  (fesQ- 

4.  23,945  -^  160  {te%t^, 

5.  302,050  -  92,059. 

6.  17  hr.  35  min.  =  how  many  minutes  ? 

7.  639,800-- 700  (te%t), 

8.  4320  square  yards  =  how  many  square  rods  ? 

9.  When  60  bu.  of  oats  grow  on  an  acre  of  ground,  how 
many  bushels  grow  on  a  square  rod  ? 

10.  The  highest  ten  batting  averages  in  the  National  Base- 
ball League  in  a  certain  year  were  .377,  .363,  .356,  .328,  .317, 
.316,   .315,   .311,   .308,   .304.     What  was  the  average   of  all 

these  ? 

11.  Using  cancellation,  divide  48  x  54  x  200  by  18  x  108  x 
25. 

12.  How  many  pieces  of  sheeting,  39  yd.  in  a  piece,  worth 
12^  a  yard,  would  pay  for  52  hours'  work  for  36  men  at  32  ^  an 
hour  ? 


162  REVIEW   AND  PRACTICE 

13.  Find  the  L.  C.  M.  of  23,  37,  32,  36,  and  56. 

14.  Find  the  G.  C.  D.  of  48,  60,  and  78. 

15.  Change  to  lowest  terms: 

16.  Reduce  to  simplest  form : 

a.  Iff.  h.  82fA.  c.  ^i-.  d.  ^^.  e.  ^.  /.  ^^. 
ff-    H^- 

17.  Change  to  improper  fractions : 

a.    399|J.     b.   48|J.     c.   18^.     d.   5&^\. 

18.  Change :  a.  7|  to  24ths.  b.  U||  to  a  fraction  whose 
terms  are  prime  to  each  other,  c.  |,  |,  and  -^^  to  fractions 
having  the  least  common  denominator. 

19.  A  wagon  which  cost  $52 J  was  sold  for  f  46|.  a.  What 
was  the  loss  ?     h.    What  per  cent  of  the  cost  was  lost  ? 

20.  A  salesman  cut  19|  yd.  of  cloth  from  a  piece  containing 
38|^  yd.     How  many  yards  remained  ? 

21.  The  remainder  is  632|^  and  the  minuend  965|.  What  is 
the  subtrahend  ? 

22.  a.  After  John  had  spent  J  of  his  money  for  a  book,  ^  of 
it  for  a  knife,  and  ^  of  it  for  oranges,  what  per  cent  of  it  was 
left  ?     b.    If  he  had  $.36  left,  how  much  had  he  at  first  ? 

23.  What  can  6  men  earn  in  4J  wk.  at  $12 J  a  week  ? 

24.  ^x8xifx6|x|f=? 

25.  A  man  sold  a  horse  for  $160  and  a  cow  for  |  as  much. 
What  did  he  receive  for  both  ? 

26.  Find  the  area  and  the  perimeter  of  a  rectangle  12^  inches 
by  8|^  inches. 

27.  Find  I  of  f  of  -^-q. 


REVIEW   AND  PRACTICE  163 

28.  A  man  sold  |  of  his  farm  and  had  90  acres  left.  How 
many  acres  did  the  farm  contain  ? 

29.  Simplify^. 

3 

30.  A  man  put  in  the  bank  f  252,  which  was  f  of  what  he 
received  for  a  wood  lot.  What  was  the  selling  price  of  the 
wood  lot  ?  • 

31.  A  merchant  lost  |  of  his  money  and  has  §123.50  left. 
How  much  had  he  at  first  ? 


32.    Simplify  ¥^. 


i  of  2i 

33.  The  sum  of  two  fractions  is  -^||.  One  of  them  is  -Jf . 
What  is  the  other  ? 

34.  Add  ten  and  five  thousandths,  three  and  seven  tenths, 
forty-seven  millionths,  five  hundred  five  thousandths. 

35.  Find  the  sum  of  eight  and  thirty-five  thousandths, 
seventeen  and  fifty-three  thousandths,  fifty  and  fifty-four 
millionths,  five  hundred  two  and  nine  ten-thousandths. 

36.  Find  the  sum  of  6.06;  70.50;  6.0765;  .00365;  101.09; 
28.56741;  50.005. 

37.  Express  in  figures  and  add :  25  thousandths,  12  hun- 
dredths, 26  ten-thousandths,  8  hundred-thousandths,  7  mil- 
lionths, 2375  hundred-thousandths. 

38.  Add  seventeen  thousandths,  eighteen  ten-thousandths, 
sixty-four  millionths,  fifteen  ten-millionths,  five  hundred  two 
hundred-thousandths,  and  from  the  sum  subtract  eighty-four 
hundred-thousandths. 

39.  a.   .0375  is  how  much  greater  than  ^^0^? 
6.  12.5-9.0025  =  ? 


164  REVIEW   AND  PRACTICE 

40.  Reduce  to  common  fractions  or  mixed  numbers: 
a.    .00125.     h.  .0875.     c,  3.625.     d.  4.032.     e.  .83J-. 

41.  Multiply  and  test  your  work : 

a.  .00375  by  400  /.  34.05  by  | 

5.  5.275  by  5000  g.  .000568  by  1.07 

c,  5.64  by  .006  h,  4.32  by  .15 

d,  35.005  by  .008  i,  '.0316  by  .58 

e,  350.5  by  8.04  j.  .375  by  2.05 

h.    Four  and  twenty  thousandths  by  twenty-six   and  nine 
tenths. 

42.  How  much  a  ton  do  I  pay  for  coal  when  .375  of  a  ton 
costs  me  81.875? 

43.  At  what  rate  per  hour  is  a  launch  running  when  it  goes 
132.3  miles  in  13.5  hours  ?  . 

44.  A  farmer  buys  groceries  and  sells  farm  produce  to  the 
grocer  as  follows : 

Groceries  Farm  Produce 

10  gal.  oil  at  2^f  25  lb.  cheese  at  18  f 

50  lb.  sugar  at  5|^  40  bu.  potatoes  at  58/ 

2  boxes  soap  at  |3,25  J  T.  hay  at  $12 

3  dozen  oranges  at  40 >^  20  doz.  eggs  at  25/ 
5  gal.  molasses  at  40/  7  bu.  pears  at  $1.50 

20  lb.  coffee  at  35  /  72  lb.  smoked  ham  at  f  .13 

In  whose  favor  is  the  balance  of  this  account,  and  how  much 
is  due  him  ? 

45.  10%  of  a  man's  income  was  paid  for  rent.     If  his  rent 
was  il5  a  month,  what  was  his  income  per  year? 

46.  After  using  70  %  of  his  month's  wages,  Jerry  had  114.40 
left.     What  were  his  month's  wages  ? 


REVIEW  AND  PRACTICE  165 

47.  a.  A  schoolroom  30  ft.  square  and  12  ft.  high  contains 
how  many  cubic  feet  of  air  ? 

h.  If  there  are  30  pupils  in  the  room,  how  many  cubic  feet 
of  air  are  there  for  each  pupil  ? 

48.  a.  Supposing  a  cubic  foot  of  ice  to  weigh  62 J  lb.,  what 
is  the  weight  of  a  pile  of  ice  12'  by  10'  by  8'  ? 

5.    How  many  three-ton  loads  would  it  make  ? 
c.   If  17|  %  of  the  ice  melts  in  handling,  how  many  pounds 
can  customers  receive  from  this  pile  of  ice  ? 

49.  Amos  sells  vegetables  for  Mr.  Robbins,  the  gardener,  and 
is  allowed  to  keep  12|  %  of  all  the  money  he  takes  in.  a.  If 
he  earned  13.41  in  a  week,  what  was  the  amount  of  his  sales  for 
that  week?     h.    How  much  did  Mr.  Robbins  receive  ? 

50.  I  bought  75%  of  a  carload  of  sugar  and  sold  |^  of  my 
share.     What  per  cent  of  the  carload  did  I  sell  ? 

51.  97%  of  the  pupils  of  a  certain  school  are  present.  If 
21  are  absent,  how  many  pupils  belong  to  the  school  ? 

52.  Three  days  are  what  per  cent  of  a  week  ? 

53.  A  piano  was  sold  for  1700.  This  was  140%  of  what  it 
cost  the  dealer,  a.  How  much  did  the  dealer  pay?  h.  How 
much  did  he  gain  ? 

54.  H.  J.  Howe,  jewelry  merchant,  sold  to  Mrs.  James  R. 
Hazzard  \  doz.  silver  table-spoons  at  $30  a  dozen,  one  dozen 
silver  table-forks  at  $25  a  dozen,  one  tea  urn  $15.50,  one 
kitchen  clock,  $2.00.     Who  is  the  debtor?     The  creditor  ? 

Make  out  the  bill  and  receipt  it. 

55.  What  is  the  amount  of  my  bill  for  4J  lb.  of  mutton 
steak  at  16^,  1^  lb.  of  tea  at  48^,  5  lb.  coffee  at  34^,  and  two 
lb.  raisins  at  15^? 


166  DENOMINATE  NUMBERS 


DENOMINATE    NUMBERS 


191.  A  number  that  is  composed  of  units  of  weight  or  measure 
is  a  denominate  number;  e,g,  10  doz.,  215  cu.  in.,  2  gal.  3  qt. 

1  pt. 

192.  The  name  of  a  unit  of  weight  or  measure  is  a  denomina- 
tion ;  e.g,  ounce,  square  foot,  minute. 

193.  A  denominate  number  that  is  expressed  in  two  or  more 
denominations  is  a   compound  number;    e.g.  1  yd.  2  ft.  7  in.; 

2  lb.  14  oz. 

194.  TABLE   OF  LIQUID  MEASURE 

4  gills  (gi.)  =  1  pint  (pt.). 
2  pints  =  1  quart  (qt.). 

4  quarts         =  1  gallon  (gal.). 

Oil,  vinegar,  molasses,  and  other  liquids  are  shipped  in  barrels 
or  casks  of  various  sizes.  But  for  the  purpose  of  indicating  the 
capacities  of  vats,  tanks,  reservoirs,  etc.,  31 J  gallons  are  called 
a,  barrel  (bbl.)  and  63  gallons  a  hogshead  (hhd.). 

195.  Oral 

1.  5  gal.  =  pt. 

2.  1  hhd.  =  bbl. 

3.  What  will  48  pt.  of  cream  cost  at  f  1.20  per  gallon? 

4.  1  bbl.  is  what  per  cent  of  1  hhd.? 
5»  How  many  pints  in  10  gal.? 

6.  At  4  ^  a  pint,  what  is  the  cost  of  6  qt.  of  milk? 

7.  4  gal.  2  qt.  1  pt.  =  pt. 


DRY  MEASURE  167 

8.  A  tank  contains  10  bbl.  of  oil.     How  many  gallon  cans 
will  it  fill? 

9.  63  qt.  is  what  part  of  a  hogshead  ? 

10.  A  gallon  contains  231  cu.  in.  How  many  cubic  inches 
are  there  in  J  of  a  gallon  ?  In  ^  of  a  gallon  ?  In  -jl^  of  a 
gallon?     In  1  qt.? 

11.  How  many  gallons  and  quarts  are  there  in  50  quarts  ? 

12.  A  cistern  that  holds  10  hhd.  of  water  holds  how  many 
barrels?     How  many  gallons? 

13.  One  pint  is  what  per  cent  of  one  gallon? 

14.  10  %  of  a  barrel  is  how  many  gallons  ? 

15.  If  1  qt.  of  sirup  can  be  made  from  20  oz.  of  maple  sugar, 
how  many  ounces  will  make  a  gallon  of  sirup  ? 

16.  33i  %  of  a  hogshead  is  how  many  gallons  ? 

196.  TABLE  OF  DRY  MEASURE 
2  pints  (pt.)  =  1  quart  (qt.). 
8  quarts         =  1  peck  (pk.). 

4  pecks  =  1  bushel  (bu.). 

197.  Oral 

1.  64  qt.  =  pk. 

2.  1  bu.  =  qt. 

3.  If  2  qt.  of  cherries  fill  a  jar,  how  many  jars  will  2  bu.  fill  ? 

4.  1  pk.  is  what  per  cent  of  a  bushel? 

5.  1  qt.  is  what  per  cent  of  a  peck? 

6.  Elsie,  Nina,  and  Robert  gathered  4  bu.  of  chestnuts  and 
sold  them  for  10^  a  quart.     How  much  did  they  receive  ? 

7.  10  bu.  = pt. 


168  AVOIRDUPOIS  WEIGHT 

8.  ^hu.  +^  pk.  = qt. 

9.  8  qt.  =  what  part  of  2  bu.  ? 

10.  What  is  gained  on  a  bushel  of  hickory  nuts  bought  for 
$2  and  sold  at  10^  a  quart  ? 

11.  A  barrel  of  potatoes  containing  2|  bushels  will  sell  for 
how  much  at  20^  a  peck? 

12.  3  bu.  and  3  pk.  of  apples  at  $1  a  bushel  cost  how  much? 

13.  If  a  bushel  of  oats  weighs  32  lb.,  what  is  the  weight  of 
3}  pk.  ? 

14.  How  many  bushels  of  apples  at  25  ^  a  peck  can  be  bought 
for  120? 

15.  A  bushel  of  corn  and  a  peck  of  wheat  are  ground  to- 
gether. What  per  cent  of  the  mixture  is  corn?  What  per 
cent  is  wheat  ? 

16.  37 J  9^  of  a  bushel  is  how  many  quarts  ? 


198,  TABLE  OF  AVOIRDUPOIS  WEIGHT 

16  ounces  (oz.)  =  1  pound  (lb.). 
2000  pounds  =  1  ton  (T.). 

2240  pounds  =  1  long  ton. 

100  pounds  =1  hundredweight  (cwt.). 

The  term  hundredweight  is  used  less  than  formerly,  although 
its  value  (100  lb.)  is  still  taken  as  a  unit  in  quoting  freight 
rates  and  prices  of  various  articles,  when  the  quantity  used 
makes  this  a  convenient  unit  of  weight. 

The  long  ton  is  used  in  wholesaling  certain  mining  products. 

The  ton  of  2000  lb.  is  sometimes  called  a  short  ton. 


LINEAR  MEASURE  169 

199.  Oral 

1.  How  many  ounces  are  there  in  1  ton  ? 

2.  At  48^  a  pound,  what  must  be  paid  for  4  oz.  of  tea  ? 

3.  1  %  of  a  ton  is  how  many  pounds  ? 

4.  1  cwt.  is  what  per  cent  of  a  ton  ? 

5.  What  is  the  cost  of  a  ton  of  corn  meal  at  f  1.25  per  hun- 
dredweight ? 

6.  How  many  short  tons  equal  1  long  ton  ? 

7.  What  is  the  cost  of  500  lb.  of  hay  at  $12  a  ton  ? 

8.  How  many  pounds  of   coal  at  $6  a  short  ton  can  be 
bought  for  11.50  ?  y 

9.  A  car  was  loaded  at  the  mines  with  10  long  tons  of  coal. 
How  many  pounds  of  coal  did  it  carry  ? 

10.  One  ounce  is  what  per  cent  of  a  pound  ? 

11.  5  lb.  of  candy  will  make  how  many  4-ounce  packages  ? 

200.  TABLE  OF  LINEAR  MEASURE 
12  inches  (in.)  =  1  foot  (ft.). 

3  feet  =  1  yard  (yd.). 

5J  yards 

or  =1  rod  (rd.). 

lejfeet 
320  rods  =  1  mile  (mL). 

201.  Oral 

1.  How  many  inches  in  10  yd.  ? 

2.  One  foot  is  what  per  cent  of  a  yard  ? 

3.  One  inch  is  what  per  cent  of  one  foot  ? 


170  LINEAR  MEASURE 

4.  How  many  rods  are  there  in  2  mi.?     In  10  mi.?     In 
100  mi.? 

5.  12  yd.  2  ft.  = ft. 

6.  33|^ %  of  2  rd.  =  how  many  feet? 

7.  10  rods  are  how  many  feet  ? 

8.  12J  %  of  a  mile  is  how  many  rods  ? 

9.  8 J  %  of  a  foot  is  how  many  inches  ?     25  %   of  a  foot  ? 
50%?     16f%?     331%? 

10.  How  many  rods  are  there  in  the  perimeter  of  a  lawn 
that  is  33  feet  square  ? 

11.  Draw  on  the  blackboard  a  line  1  yd.  long,  using  no 
measure.  Measure  and  correct  it.  On  your  paper  draw  a  line 
3 1  in.  long,  using  no  measure.     Measure  and  correct  it. 

12.  Estimate  the  length  and  breadth  of  your  schoolroom. 
Test  your  estimates  by  measuring. 

13.  How  many  feet  high  do  yo\i  think  your  schoolroom  is  ? 
Can  you  find  a  way  to  measure  it  without  climbing  ?  Measure 
it,  and  see  how  nearly  correct  your  estimate  is. 

14. 


This  oblong  represents  a  field.  ^  inch  stands 
for  1  rod.  Measure  the  sides,  and  tell  how  many 
rods  long  and  wide  the  field  is. 


15.  Draw  on  the  blackboard  a  plan  of  your  schoolroom  floor, 
using  ^  inch  for  1  foot ;  that  is,  draw  the  floor  to  the  scale  of 
1'  to  J". 


SUKFACE  MEASURE 


171 


16.  This  square  represents  a  square 
mile.     Can  you  tell  what  the  scale  is  ? 

Each  small  square  is  what  part  of  a 
square  mile? 

How  many  rods  of  fence  would  be 
needed  to  inclose  a  square  field  con- 
taining ^  of  a  square  mile  ? 

17. 


320  rd. 


The  width  of  this  floor  is  16  ft. 
Can  you  find  the  scale  of  this  plan  ? 

What  is  the  length  of  one  long  side? 
What  is   the    greatest    length    of    the 
room? 


18.  If  you  go  30  inches  at  a  step,  how  many  steps  will  you 
take  in  going  30  feet? 

19.  If  a  row  of  corn  contains  five  hills  to  a  rod,  how  many 
hills  are  there  in  a  row  a  quarter  of  a  mile  long? 

202.        TABLE  OF  SURFACE  MEASURE 

144  square  inches  (sq.in.)  =  1  square  foot  (sq.  ft.). 

9  square  feet  =  1  square  yard  (sq.  yd.). 

30 J  square  yards  =  1  square  rod  (sq.  rd.). 

160  square  rods  =  1  acre  (A.). 

640  acres  =1  square  mile  (sq.  mi.). 


172  SURFACE  MEASURE 

203.    Oral 

1.  Without  a  measure  draw  a  square  inch.  Measure  and 
correct  it. 

2.  Without  a  measure  draw  on  the  blackboard  a  square  foot 
and  a  square  yard.  Measure  and  correct  them.  Divide  the 
square  yard  into  square  feet.  Divide  the  square  foot  into 
square  inches. 

3.  How  many  square  inches  are  there  in  two  square  feet  ? 

4.  Estimate  the  number  of  square  yards  in  the  floor  of  your 
schoolroom.  Measure  it,  and  see  how  nearly  right  your  esti- 
mate is.  Make  an  estimate  of  the  area  of  each  wall,  and  test 
it  by  measuring.  Measure  your  school  lot,  and  find  what  part 
of  an  acre  it  contains. 

5.  How  many  square  inches  are  there  in  -|-  sq.  ft.  ? 

6.  One  square  yard  is  ^  of  how  many  square  feet  ? 

7.  A  5-inch  square  contains  how  many  square  inches  ? 
Draw  it. 

8.  A  rectangle  6''  by  12"  is  what  part  of  a  square  foot  ? 

9.  How  many  tiles  6''  square  will  cover  a  floor  10  ft.  by 
5  ft.? 

10.  8  sq.  in.  are  what  part  of  an  8-inch  square  ? 

11.  How  many  square  yards  are  there  in  4  sq.  rd.  ? 

12.  40  sq.  rd.  are  what  per  cent  of  an  acre  ? 

13.  A  room  is  15  ft.  by  12  ft.  and  9  ft.  high.  How  many 
square  yards  are  there  in  the  floor  ?  Draw  a  plan  of  it  to  the 
scale  of  y  =  1'. 

How  many  square  yards  are  there  in  one  long  wall  ?  In  one 
short  wall  ?     Draw  a  plan  of  each  wall. 


VOLUME  MEASURE 


173 


How  many  square  yards  of  plastering  are  needed   for  the 
ceiling  ? 

14.  How  many  acres  are  there  in  a  farm  160  rd.  long  and 
100  rd.  wide  ? 

15.  How  wide  must  a  field  be  to  contain  10  A.  if  it  is  40  rd. 
long  ? 

16.  A  10-acre  field  is  20  rd.  wide.     How  long  is  it  ? 


B 

A 

17.    The  scale  of  these  plans  is  16'  to  1''.    Find  the  perimeter 
and  area  of  the  surface  represented  by  each. 


204.  TABLE  OF  VOLUME  MEASURE 

1728  cubic  inches  (cu.  in.)  =  1  cubic  foot  (cu.  ft.). 
27  cubic  feet  =  1  cubic  yard  (cu.  yd.). 

205.  Oral 

1.  Define  a  cube. 

2.  How  many  edges  has  a  cube  ?     How  do  they  compare  ? 

3.  How  many  cubic  inches  are  there  in  a  6-inch  cube  ? 


174  TABLE  OF  TIME 

4.  Prove  that  a  tight  tin  box  7  in.  by  11  in.  by  3  in.  will 
hold  one  gallon. 

5.  A  block  of  wood  4  in.  square  must  be  how  long  to  con- 
tain 96  cu.  in.  ? 

6.  18  cu.  ft.  are  what  part  of  a  cubic  yard  ? 

7.  A  box  12  ft.  long  and  3  ft.  wide  must  be  how  deep  to  hold 
4  cu.  yd.  of  sand  ? 

8.  8 J  %  of  a  cubic  foot  is  how  many  cubic  inches  ? 

9.  A  2-inch  cube  is  equal  to  how  many  1-inch  cubes  ? 
10.    How  many  2-inch  cubes  would  make  a  4-inch  cube  ? 

206.  TABLE  OF  TIME 

60  seconds  (sec.)  =  1  minute  (rain.). 
60  minutes  =  1  hour  (hr.). 

24  hours  =  1  day  (da.). 

7  days  =  1  week  (wk.). 

365  days  =  1  common  year  (yr.). 

366  days  =  1  leap  year. 

Ten  years  are  called  a  decade^  and  one  hundred  years  make 
a  century^  but  these  terms  are  not  used  in  arithmetical  cal- 
culations. 

The  four  thirty-day  months  may  be  remembered  easily  by 
the  following  old  rhyme  : 

"  Thirty  days  hath  September, 
April,  June,  and  November." 

February  has  28  days,  with  29  in  leap  year.  The  other 
months  have  31  days. 


TABLE  OF  COUNTING  175 

207.  Oral 

1.  How  many  minutes  are  there  in  a  working  day  of  8  hr.  ? 

2.  A  man  who  works  for  30  ^  an  hour  receives  how  much  a 
minute  ? 

3.  A  train  that  is  running  at  the  rate  of  2  miles  in  3 
minutes  goes  how  many  miles  in  an  hour?  In  10  hours? 
In  24  hours? 

4.  A  boy  who  is  idle  15  minutes  in  every  hour  wastes  what 
per  cent  of  his  time  ? 

5.  How  many  hours  are  there  in  a  week  ?  In  the  month  of 
June  ? 

6.  How  many  hours  have  we  for  work  in  a  morning  session 
of  school  if  it  begins  at  9  o'clock  and  closes  at  11.45,  allowing 
a  quarter  of  an  hour  for  recess  ? 

7.  How  many  days  are  there  in  the  fall  months? 

8.  Close  your  book.  Recite  the  table  of  time  and  the  names 
of  the  months,  giving  the  number  of  days  in  each  month. 

208.  TABLE  OF  COUNTING 
12  =1  dozen  (doz.). 
12  doz.  =  1  gross. 

20  =1  score. 

209.  Oral 

1.  How  much  apiece  do  oranges  cost  at  40^  a  dozen? 

2.  One  dozen  is  what  per  cent  of  1  score  ?     Of  1  gross? 

3.  How  many  pens  are  there  in  a  gross  ? 

4.  If  I  buy  pens  at  72  ^  a  gross  and  sell  them  at  1  f^  apiece, 
how  much  do  I  make  on  a  gross  ?  On  a  dozen  ?  On  a  pen  ? 
I  gain  what  per  cent  of  the  cost  ? 


176  PAPER  MEASURE 

5.  "Fourscore  and  seven  years  ago"  was  how  many  years 
Ago? 

6.  A  merchant  bought  fiber  pails  at  f  3  a  dozen.  How 
much  apiece  did  he  pay  ?  If  he  sold  them  at  35  ^  apiece,  what 
did  he  gain  on  one  ?  On  a  gross  ?  What  per  cent  of  the  cost 
did  he  gain  ? 

7.  A  merchant  buys  shoe  brushes  at  11.20  a  dozen  and  sells 
them  at  15  ^  apiece.  How  much  does  he  gain  on  one  ?  What 
per  cent  of  the  cost  does  he  gain  ? 

8.  What  is  the  cost  of  6  cans  of  Alaska  salmon  at  98  ^  a 
dozen  cans  ? 

9.  What  is  the  cost  of  a  gross  of  pencils  at  40^  a  dozen? 

10.  A  man's  age  is  threescore  and  ten  years.  How  many 
years  old  is  he  ? 

11.  Bars  of  soap  at  1 9  a  gross  are  how  much  a  dozen  ? 

210.  TABLE   OF  PAPER  MEASURE 

24  sheets  =  1  quire. 
20  quires  =  1  ream. 

The  terms  bundle  (2  reams)  and  hale  (5  bundles)  are  seldom 
used.  The  denomination  quire  is  used  mostly  in  measuring  the 
finer  grades  of  writing  paper.  Wrapping  paper  is  sold  by  the 
pound  or  by  the  thousand  sheets.  Many  kinds  of  paper  are 
sold  in  packages  of  five  hundred  or  one  thousand  sheets.  Pack- 
ages of  five  hundred  sheets  are  sometimes  called  reams, 

211.  Oral 

1.  How  many  sheets  of  paper  are  there  in  2  quires  ?  In  4 
quires  ?  In  ^  of  a  ream  ?  In  3  quires  ?  In  |^  ream  ?  In 
10  reams  ? 


ARC   AND  ANGLE  MEASURE  177 

2.  One  quire  is  what  per  cent  of  1  ream  ?  Of  ^  ream  ?  Of 
•|  ream  ? 

3.  What  is  the  profit  on  10  quires  of  paper  bought  at  14 
cents  a  quire  and  sold  at  a  cent  a  sheet  ? 

4.  A  package  of  500  sheets  of  paper  contains  how  much  more 
than  twenty  quires  ? 

5.  A  stationer  sold  10  quires  out  of  a  package  of  1000  sheets 
of  paper.  How  many  sheets  were  left  ?  What  per  cent  of 
the  package  was  sold  ?    What  per  cent  was  left  ? 

6.  A  stationer  made  a  dozen  tablets,  each  containing  72 
sheets  of  paper.     How  many  quires  were  used  for  each  tablet? 

The  paper  cost  40^  a  ream.  What  was  the  cost  per  quire  ? 
What  was  the  cost  of  the  paper  in  one  tablet  ?  What  was  the 
cost  of  the  paper  for  a  dozen  tablets  ? 

If  the  backs  and  labor  cost  28^  for  a  dozen  tablets,  what 
was  the  entire  cost  of  a  dozen  tablets  ? 

If  they  were  sold  for  10^  a  piece,  what  was  the  gain  on  a 
dozen  tablets  ?  The  gain  was  what  per  cent  of  the  cost  ? 
What  was  the  gain  on  a  gross  of  tablets  ? 

7.  One  quire  of  paper  will  make  how  many  leaves  if  each 
sheet  is  folded  into  8  leaves  ? 

8.  If  12  sheets  of  a  certain  kind  of  paper  weigh  one  pound, 
how  many  pounds  will  5  quires  weigh  ? 

9.  960  pages  in  a  book  would  require  how  many  leaves  ?  If 
one  sheet  makes  4  leaves,  how  many  sheets  are  required  ? 
How  many  quires  ? 

212.  TABLE  OF  ARC   AND  ANGLE  MEASURE 

60  seconds  (")  =1  minute  (')• 
60  minutes  =  1  degree  (°). 
An  arc  of  360°  =  1  circumference. 


178 


ARC   AND  ANGLE  MEASURE 


213.    The  difference  in  direction  of  two 
angle  ;  e.g. 


that  meet  is  an 


214.  The  lines  that  meet  to  form  an  angle 
are  the  sides  of  the  angle. 

Lines  are  read  by  means  of  letters  placed 
at  their  extremities.  Angles  are  read  by 
means  of  letters  placed  at  the  extremities  of  their  sides. 

In  the  angle  ABQ  the  lines  AB  and  BO  are  the  sides. 

215.  The  sum  of  all  the  angles  that  can  he  formed  around  a 
point  in  a  plane  is  360°, 


120 


120 


In  figure  1  there  are  three  angles  about  a 
point.     Add  the  numbers  of  degrees. 


Fig.  1 


In  figure   2  there    are  five  angles  about  a 
point.     Add  the  numbers  of  degrees. 


90° 


90' 


90' 


90' 


Fia.  3 


Fig.  2 


In  figure  3  there  are  four  angles  about  a 
point.     Add  the  numbers  of  degrees. 

Draw  eight  equal  angles  about  a  point. 
How   many   degrees    are    there   in   each 
angle  ? 

Make  other  questions  about  these  angles. 


ARC   AND  ANGLE  MEASURE  179 

216.     Oral 

1.  How  many  angles  are  there  in  Fig.  3,  page  178  ?  How  do 
they  compare  ?  Each  of  these  angles  is  a  right  angle.  How 
many  degrees  are  there  in  a  right  angle  ? 

2.  When  the  hour  hand  of  the  clock  is  at  12  and  the  minute 
hand  is  at  9,  they  form  what  kind  of  an  angle  ?  At  what  other 
number  could  the  minute  hand  point  to  make  a  right  angle 
with  the  hour  hand  at  12  ? 

An  angle  of  90°  is  a  right  angle. 

An  angle  that  is  greater  than  a  right  angle  is  an       , 
obtuse  angle. 

An  angle  that  is  less  than  a  right  angle  is  an 
acute  angle. 


3.  Draw  a  right  angle.     How  many  degrees  are 

there  in  it?     Divide  it  in  the  middle  by  a  line.     How  many 
degrees  are  there  in  each  of  the  angles  thus  formed  ? 

4.  Make  a  drawing  of  a  wagon  wheel  with  6  spokes.  The 
spokes  form  angles  of  how  many  degrees  ?  Put  in  twice  as 
many  spokes.  How  many  degrees  are  there  in  the  angles  ? 
Double  the  number  again,  and  tell  the  size  of  the  angles. 
What  kind  of  angles  are  these  ? 

5.  Draw  an  angle  that  you  think  is  about  an  angle  of  one 
degree. 

6.  One  minute  is  what  part  of  a  degree  ?  Can  you  think  of 
something  that  is  like  an  angle  of  one  minute  ? 

7.  The  minute  hand  of  a  clock  passes  through  how  many 
degrees  in  12  hours  ?     In  1  hour  ? 

8.  What  kind  of  angle  (right,  obtuse,  or  acute)  is  formed 
by  the  hour  and  minute  hands  of  a  clock  at  two  o'clock  ?  At 
five  o'clock  ?     At  eleven  o'clock  ? 


180 


ARC   AND  ANGLE  MEASURE 


9.    10°  =  how  many  minutes  ?     20°  ?     40°  ?     i°  ?     J/  ? 

10.  1°  =  how  many  seconds  ? 

11.  Stand  facing  north.  Turn  90°  to  the  left.  In  what  di- 
rection are  you  facing  ?  Turn  90°  farther.  In  what  direction 
are  you  facing?  Turn  180°  farther.  In  what  direction  are 
you  facing  ?     How  many  degrees  have  you  turned  in  all  ? 

217.  A  plane  figure  hounded  hy  a  curved  line^  every  point  of 
which  is  equally  distant  from  a  point  within,  called  the  center,  is 
a  circle. 

218.  The  boundary  line  of  a  circle  is  its  circumference. 

219.  Any  part  of  a  circumference  is  an  arc. 

The  number  of  degrees  in  an  arc  is  always  the  same  as  the 
number  of  degrees  in  the  angle  at  the  center  whose  sides  meet  the 
extremities  of  the  arc,  thus : 

The  angle  AOB  is  -J  the  sum  of  all  the 
angles  at  the  center,  or  90°.  The  arc  AB 
is  \  of  the  circumference,  or  90°.  Can  you 
tell  the  number  of  degrees  in  the  arc  BQl 
In  the  angle  BOQl 

Note.  —  The  number  of  degrees  in  any  angle  may  be  measured  by 
means  of  a  protractor,  an  instrument  with  the  degrees  marked  and  num- 
bered. 


A  Protractor 


UNITED   STATES  MONEY  181 

220.  TABLE   OF   UNITED  STATES  MONEY 

10  mills  =  1  cent. 
10  cents  =  1  dime. 
10  dimes  =  1  dollar. 

The  gold  coins  of  the  United  States  are  the  85,  f  10,  and 
$20  pieces,  once  called  the  half  eagle,  eagle,  and  double  eagle. 
Gold  dollars  are  not  in  general  circulation,  although  a  few  of 
them  have  been  coined. 

The  silver  coins  are  the  dollar,  half  dollar,  quarter  dollar, 
and  dime.  Silver  half-dimes  are  no  longer  coined.  Most  five- 
cent  pieces  are  made  of  nickel.  Most  1-cent  pieces  are  made 
of  bronze,  though  some  nickel  and  copper  cents  are  in 
circulation. 

The  mill  is  not  coined. 

221.  Oral 

1.  One  dime  is  what  part  of  a  dollar  ?     What  per  cent  ? 

2.  One  cent  is  what  per  cent  of  a  dollar  ? 

3.  One  mill  is  what  part  of  a  dollar  ?     What  per  cent  ? 

4.  79  cents  is  what  decimal  of  a  dollar?  7  mills  is  what 
decimal  of  a  dollar  ?     19  cents  and  7  mills  ? 

5.  Express  as  decimals  of  a  dollar : 

85  cents  6  mills;  10  cents  8  mills  ;  4  cents  7  mills;  8  cents ; 
29  cents  1  mill ;  3  mills. 

6.  5  mills  are  what  part  of  a  cent  ?     4  mills  ?     8  mills  ? 

7.  The  value  of  a  $5  gold  piece  is  what  per  cent  of  the 
value  of  a  110  gold  piece  ?     Of  a  §20  gold  piece  ? 

8.  Make  other  problems  about  dollars,  mills,  and  cents. 


182  TROY  AND  APOTHECARIES'  WEIGHTS 

222.  TABLE  OF  TROY  WEIGHT 

24  grains  (gr.)      =  1  pennyweight  (pwt.). 
20  pennyweights  =  1  ounce  (oz.). 
12  ounces  =  1  pound  (lb.). 

These  weights  are  used  in  weighing  gold,  silver,  and  some 
jewels.  To  get  an  idea  of  the  weight  of  a  grain,  think  of  the 
weight  of  a  grain  of  wheat  or  rice. 

223.  Oral 

1.  How  many  grains  are  there  in  1  Troy  ounce  ? 

2.  A  silver  dollar  weighs  about  412|  grains.  This  is  how 
much  less  than  a  Troy  ounce  ? 

3.  A  gold  dollar  contains  23.2  grains  of  pure  gold,  but 
enough  harder  metal  is  put  with  the  gold  to  make  it  weigh  25.8 
grains.     This  weight  is  how  much  more  than  1  pwt.  ? 

4.  Calling  the  weight  of  a  gold  dollar  1  pwt.,  what  does  a  $20 
gold  piece  weigh?  How  many  dollars  in  gold  would  weigh 
a  pound  ? 

5.  How  many  Troy  ounces  would  $1000  in  gold  weigh? 
How  many  Troy  pounds  ? 

6.  What  must  I  pay  for  a  watch  chain  weighing  240  grains 
at  $1  a  pennyweight  ? 

224.  TABLE  OF  APOTHECARIES'   WEIGHT 
20  grains  (gr.)  =  1  scruple  (sc.  or  3). 

3  scruples  =  1  dram  (dr.  or  3). 

8  drams  =  1  ounce  (oz.  or  5). 

This  table  is  used  by  druggists  and  physicians  in  compound- 
ing medicines  ;  but  medicines  are  bought  and  sold  by  avoir- 
dupois weight,  except  in  quantities  smaller  than  one  ounce. 


REDUCTION  OF  DENOMINATE  NUMBERS  183 

Druggists  also  use  a  term  fluid  ounce^  which  is  not  a  measure 
of  weight,  but  of  capacity,  and  is  equal  to  -^^  of  a  pint.  Thus, 
a  2-ounce  bottle  is  a  bottle  that  holds  ^  of  a  pint  of  any  liquid 
regardless  of  its  weight. 

225.  Oral 

1.  A  druggist  buys  a  pound  (avoirdupois)  of  quinine  con- 
taining 7000  grains.  How  many  2-grain  tablets  can  be  made 
from  it  ? 

2.  How  many  2-grain  tablets  can  be  made  from  1  3  ? 

3.  How  many  3-grain  tablets  can  be  made  from  1  3  ? 

4.  A  patient  takes  5  gr.  of  a  certain  medicine  every  day. 
How  long  will  1  3  of  it  last  him  ? 

5.  A  druggist  made  1000  powders,  each  containing  2  gr.  of 
ipecac,  2  gr.  of  muriate  of  ammonia,  and  10  gr.  of  extract  of 
licorice.  There  are  7000  grains  in  1  lb.  avoirdupois.  How 
many  avoirdupois  pounds  did  all  the  powders  weigh  ? 

REDUCTION  OF  DENOMINATE  NUMBERS 

226.  Changing  numbers  to  larger  denominations  is  reduction 
ascending. 

227.  Changing  numbers  to  smaller  denominations  is  reduction 
descending. 

228.  Oral 

1.  How  many  gallons  are  there  in  72  pints  ?  What  kind  of 
reduction  is  this  ? 

2.  How  many  minutes  are  there  in  10  °  ?  What  kind  of  re- 
duction is  this  ? 


184 


REDUCTION  OF  DENOMINATE  NUMBERS 


3. 

Reduce : 

a. 

2  bu.  to  quarts. 

I. 

h. 

64  pt.  to  pecks. 

m. 

c. 

17  T.  to  pounds. 

n. 

d. 

96  oz.  to  pounds. 

0, 

e. 

11  yd.  to  rods. 

P- 

/. 

33  ft.  to  rods. 

?• 

9- 

5  A.  to  square  rods. 

r. 

h. 

288  sq.  in.  to  square  feet. 

s. 

i. 

1728  cu.  in.  to  cubic  feet. 

t. 

h 

10  cu.  yd.  to  cubic  feet. 

u. 

k. 

20  wk.  to  days. 

V. 

^  da.  to  hours. 
^  yr.  to  days. 
J  niin.  to  seconds. 
20  da.  to  hours. 
240  sec.  to  minutes. 
96  doz.  to  gross. 
12  score  to  units. 
100  quires  to  reams. 
50  reams  to  quires. 
7'  to  seconds. 
720"  to  minutes. 


4.  Reduce : 

a,  51  qt.  to  gallons  and  quarts. 

h.  7  gal.  2  qt.  to  quarts  ;  to  pints. 

c,  35  qt.  to  bushels  and  quarts. 

d,  1  bu.  3  pk.  to  pecks  ;  to  quarts  ;  to  pints. 

e,  1  T.  370  lb.  to  pounds. 

/.  40  oz.  to  pounds  and  ounces. 

g.  15  cwt.  50  lb.  to  pounds. 

h,  1  A.  40  sq.  rd.  to  square  rods. 

i,  4  sq.  yd.  to  square  feet. 

j,  100  sq.  ft.  to  square  yards  and  square  feet, 

k.  64  fluid  oz.  to  pints. 

I,  130  min.  to  hours  and  minutes. 


REDUCTION  DESCENDING 

229.  Compound  numbers  are  seldom  expressed  in  more  than 
two  denominations.  In  measures  of  time  and  arcs  three  denom- 
inations are  sometimes  used. 

Long  and  difficult  reductions  are  seldom  necessary. 


KEDUCTION  OF  DENOMINATE  NUMBERS  185 

Reduce  17  da.  10  hr.  40  min.  to  minutes. 

17    da. 

24    number  of  hours  in  1  da. 

68 
34 

408    number  of  hours  in  17  da. 
10 

418    number  of  hours  in  17  da.  10  hr. 
60  number  of  minutes  in  1  hr. 


25080  number  of  minutes  in  418  hr. 
40 


25120  number  of  minutes  in  418  hr.  40  min., 
or  17  da.  10  hr.  40  min. 

Reduce  41  A.  20  sq.  rd.  to  square  feet. 

41        A. 

160      number  of  square  rods  in  1  A. 

2460 
41 


6560      number  of  square  rods  in  41  A. 
20 


6580  number  of  square  rods  in  41  A.  20  sq.  rd. 

30J  number  of  square  yards  in  1  sq.  rd. 

1645  (6580x1) 

197400  (6580  x  30) 

199045  number  of  square  yards  in  6580  sq.  rd. 

9  number  of  square  feet  in  1  sq.  yd. 


1791405      number  of  square  feet  in  41  A.  20  sq.  rd. 


186  REDUCTION  OF  DENOMINATE  NUMBERS 

230.  Written 
Reduce : 

1.  49  da.  7  hr.  to  hours. 

2.  79  A.  50  sq.  rd.  to  square  rods. 

3.  16  yr.  7  mo.  20  da.  to  days  (allow  30  da.  for  1  mo.). 

4.  9  sq.  yd.  6  sq.  ft.  to  square  inches. 

5.  78  T.  16  cwt.  to  hundredweight. 

6.  59  cwt.  23  lb.  to  pounds. 

7.  48  lb.  15  oz.  Troy  to  ounces. 

8.  12  cu.  ft.  384  cu.  in.  to  cubic  inches. 

9.  78  cu.  yd.  19  cu.  ft.  to  cubic  feet. 

10.  48  gal.  2  qt.  to  pints. 

11.  15  bu.  3  pk.  to  pints. 

12.  25°  30'  15^'  to  seconds. 

13.  The  degrees  in  1  right  angle  to  seconds. 

14.  158  gross  5  doz.  to  dozen. 

15.  3  mi.  to  feet. 

16.  1  mi.  to  inches. 

17.  1  sq.  mi.  to  square  rods. 

18.  21  T.  362  lb.  to  pounds. 

19.  4  cu.  yd.  to  cubic  inches. 

20.  121  yr.  11  mo.  to  months. 

21.  8  mo.  28  da.  to  days. 

22.  42  wk.  3  da.  to  hours. 

23.  17  cwt.  to  ounces. 

24.  12  yd.  19  in.  to  inches. 

25.  2  yr.  4  mo.  3  da.  to  days. 


REDUCTION   OF  DENOMINATE  NUMBERS  187 

26.  3  yr.  6  mo.  15  da.  to  days. 

27.  17°  3'  to  minutes. 

28.  7  yr.  14  da.  to  days. 

29.  15  mo.  29  da.  to  days. 

30.  42°  12'  to  seconds. 

REDUCTION  ASCENDING 
231.    Reduce  3876  sec.  to  hours,  minutes,  and  seconds. 


How  many  seconds  =  1  min.  ? 

387p  sec. 3876  sec.  =  how  many  minutes  and 

6^  min.  +  36  sec.  seconds ? 

1  hr.  +  4  min.  How  many  minutes  =  1  hr.? 

1  hr.  4  min.  36  sec.     Ans.  ^^  "^^^-  =  ^^^  ^^^^  ^^^^  ^"^ 

minutes? 


6JZ) 


Written 
Reduce : 

1.  42,876  sec.  to  hours,  minutes,  and  seconds. 

2.  16,307^'  to  degrees,  minutes,  and  seconds. 

3.  8370  da.  to  years  and  days. 

4.  983  pk.  to  bushels  and  pecks. 

5.  5834  lb.  to  tons  and  pounds. 

6.  4376  oz.  Troy  to  pounds  and  ounces. 

7.  892  pt.  to  gallons  and  quarts. 

8.  508  pt.  to  gallons  and  quarts. 

9.  8376'  to  degrees  and  minutes. 

10.  45,360"  to  degrees  and  minutes. 

11.  4416  sheets  to  quires. 

12.  685  sq.  ft.  to  square  yards  and  square  feet. 

13.  28,347  cu.  ft.  to  cubic  yards  and  cubic  feet. 


188  REDUCTION  OF  DENOMINATE  NUMBERS 

14.  38,627  sec.  to  hours,  minutes,  and  seconds. 

15.  497'  to  degrees  and  minutes. 

16.  89,764  lb.  to  tons  and  pounds. 

17.  49,763  ft.  to  miles  and  feet. 

18.  42,374  da.  to  years  and  days. 

19.  94,276  min.  to  days,  hours,  and  minutes. 

20.  13,794  ft.  to  rods. 

VARIOUS  FORMS  OF  REDUCTION 
232.    Written 

1.  How  many  inches  are   there   in  |-  rd.  ?     (Indicate   the 
work  thus  :  f  x-^^x-^^-;  then  cancel.) 

2.  Find  the  number  of: 


a. 

Inches  in  f  mi. 

/. 

Coat-hooks  in  ||^  gross. 

6, 

Pints  in  ^g  bu. 

9- 

Sheets  in  l|  ream. 

c. 

Pints  in  f  bbl. 

h. 

Cubic  inches  in  g|^g  cu.  yd, 

d. 

Seconds  in  l\'' . 

.    i. 

Minutes  in  -^^^  wk. 

e. 

Ounces  in  -^^  T. 

J- 

Square  feet  in  -^^  A. 

3.   What  part  of  5  gal.  is  1  gal.  1  pt.  ? 

Note.  —  Find  the  number  of  pints  in  5  gal. ;  then  in  1  gal.  1  pt. 
Statement  of  Relation  :  of  40  pt.  =  9  pt. 


4. 

What  part: 

a. 

Of  1  T.  is  324  lb.? 

/. 

Of  7  cu.  yd.  is  5  cu.  yd.  9 

h. 

Of  2  T.  is  7  cwt.  40  lb.  ? 

cu.  ft.  ? 

c. 

Of  3  gal.  is  2  qt.  1  pt.  ? 

9- 

Of  2  yr.  is  1  yr.  3  mo.  ? 

d. 

Of   5   da.  is    12   hr.   30 

h. 

Of  a  square  mile  is  200 

min.  ? 

A.  40  sq.  rd.  ? 

e. 

Of    a    circumference    is 

i. 

Of  20  gal.  is  5  gal.  2  qt.  ? 

36°  45'? 

J- 

Of  a  week  is  18  hr.  8  min .  ? 

REDUCTION   OF  DENOMINATE  NUMBERS  189 

5.  3  bu.  2  pk.  of  potatoes  are  sold  out  of  a  load  of  28  bu. 
a.  What  part  of  the  load  is  left?  5.  What  per  cent  of  the 
load  is  sold  ? 

6.  2  bbl.  of  cranberries,  each  containing  3  bu.,  are  worth 
how  much  at  12 J  ^  per  quart  ? 

7.  The  speed  limit  for  automobiles  in  a  certain  town  is  8 
miles  per  hour.     That  is  how  many  rods  per  minute  ? 

8.  What  is  the  cost  of  5  T.  500  lb.  of  coal  at  15.40  per  ton? 

9.  \  T.  is  equal  to  how  many  ounces  ? 

10.  How  long  will  16  bu.  of  corn  last  Fred's  chickens  if  he 
feeds  them  1  qt.  a  day  ? 

11.  a.  What  is  the  profit  on  a  bushel  of  chestnuts  bought 
for  82^  and  sold  at  10  cents  a  pint  ?  h.  The  gain  is  what  per 
cent  of  the  cost  ? 

12.  How  many  minutes  are  there  in  April,  May,  and  June  ? 

13.  What  is  the  cost  of  the  milk  supply  for  September  of 
a  housekeeper  who  buys  \  gal.  per  day  and  pays  6|^^  per 
quart  ? 

14.  What  are  the  yearly  wages  of  a  man  who  earns  a  cent 
in  2  minutes  and  works  8  hours  a  day  and  26  days  in  a  month 
throughout  the  year  ? 

15.  If  a  horse  eats  12  lb.  of  hay  per  day,  how  many  tons  and 
pounds  will  he  eat  in  a  year  ? 

16.  A  box  6''  X  4''  X  2''  contains  what  fraction  of  a  cubic 
foot  ? 

17.  I  of  an  acre  of  land  contains  how  many  square  feet  ? 

18.  16  sq.  rd.  11  sq.  yd.  are  what  part  of  an  acre  ? 


Lb. 

Oz. 

7 

8 

15 

14 

23 

15 

47 

5 

Add: 

190  ADDITION  OF   COMPOUND  NUMBERS 

ADDITION   AND    SUBTRACTION   OF   COMPOUND   NUMBERS 

233.    Written 

Add  7  lb.  8  oz.,  15  lb.  14  oz.,  23  lb.  15  oz. 

15  oz.  +  14  oz.  +  8  oz.  =  37  oz.  =  2  lb.  5  oz. 
2  lb.  +  23  lb.  +  15  lb.  +  7  lb.  =  47  lb. 
47  lb.  5  oz.    Ans. 


1.  19  ft.  6  in.,  17  ft.  10  in.,  9  ft.  6  in. 

2.  13  A.  17  sq.  rd.,  19  A.  153  sq.  rd. 

3.  2  hr.  5  min.  30  sec,  8  hr.  53  min.  47  sec. 

4.  8  gal.  3  qt.,  15  gal.  1  qt.,  16  gal.  2  qt. 

5.  81°  19'  35'',  2°  50'  29",  3°  4'  50". 

6.  6  T.  480  lb.,  7  T.  730  lb.,  19  T.  900  lb. 

7.  5  yd.  2  ft.,  16  yd.  1  ft.,  18  yd.  2  ft. 

8.  6  pk.  7  qt.,  3  pk.  5  qt.,  2  pk.  6  qt. 

9.  7  cu.  yd.  18  cu.  ft.,  12  cu.  yd.  19  cu.  ft. 

10.  6  yr.  7  mo.  3  da.,  7  yr.  8  mo.  29  da. 

11.  8  lb.  7  oz.,  16  lb.  14  oz.,  19  lb.  10  oz. 

12.  21  bu.  3  pk.,  9  bu.  2  pk.,  35  bu.  1  pk. 

13.  2  wk.  3  da.,  19  wk.  1  da.,  20  wk.  6  da. 

14.  7  hr.  38  min.  21  sec,  5  hr.  47  min.  29  sec 

15.  5  yr-  200  da.,  7  yr.  321  da.,  8  yr.  179  da. 


SUBTRACTION  OF  COMPOUND  NUMBERS  191 

234.    Written 

From  18  yr.    7  mo.  14  da. 
take       6  yr.    8  mo.  26  da. 

11  yr.  10  mo.  18  da.     Difference 

7  mo.  14  da.  =  6  mo.  44  da.  „^-      ,  ,      ,  ,      .       « 

1Q        a  1H.        1Q  (Why  do  we  make  these  reductions?) 

18  yr.  6  mo.  =  17  yr.  18  mo.  \       j  j 

18  yr.  7  mo.  14  da.  =  17  yr.  18  mo.  44  da. 

17  yr.  18  mo.  44  da.  -  6  yr.  8  mo.  26  da.  =  11  yr.  10  mo.  18  da. 

Suhtraet: 

1.    12  yr.  3  mo.  15  da.  5.    4  yr.  2  mo.  18  da. 

10  yr.  1  mo.  19  da.  1  yr.   7  mo.  12  da. 


2. 

9yr. 
3yr. 

2  mo. 
5  mo. 

Ida. 
29  da. 

3. 

9  yr. 
4  yr. 

2  mo. 
8  mo. 

5  da. 
5  da. 

4. 

12  yr. 

8  yr. 

2  mo. 

16  da. 
12  da. 

6.    7yr. 
3  yr. 

5  mo. 

6  mo. 

18  da. 
7  da. 

7.    9  yr. 

8yr. 

6  mo. 
5  mo. 

13  da. 
15  da. 

8.    1  yr. 

3  mo. 

7  mo. 

15  da. 

•        9.    How   many   years,   months,   and   days    are    there    from 
May  30,  1907,  to  Dec.  5,  1909  ? 

1909  yr.  12  mo.  5  da.  Note.  —  December  is  the  twelfth  month 
1907  vr.  5  mo.  30  da.  ^^^  ^^^  *^®  fifth.  Count  30  da.  for  a 
' month. 

Find  the  time  from  : 

10.  July  19,  1827,  to  Mar.  26,  1878, 

11.  Sept.  20,  1831,  to  Nov,  15,  1909. 


192  SUBTRACTION  OF  COMPOUND  NUMBERS 

12.  Jan.  7,  1840,  to  Feb.  8,  1896. 

13.  May  4,  1850,  to  Jan.  12,  1861. 

14.  Oct.  19,  1760,  to  Aug.  20,  1860, 

15.  Dec.  12,  1880,  to  June  5,  1903. 

16.  July  10,  1809,  to  Oct.  2,  1893. 

17.  May  8,  1899,  to  Feb.  12,  1908. 

18.  Feb.  12,  1901,  to  Jan.  30,  1906. 
Subtract : 

19.  5  hr.  54  min.  30  sec. 
1  hr.  50  min.  50  sec. 

20.  122°  31'  15'' 

60°  20'  45" 

21.  23  hr.  54  min.  36  sec. 
20  hr.  24  min.  48  sec. 

22.  7  ft.    6  in. 
3  ft.  11  in. 


23. 

27  gal. 
18  gal. 

2  qt. 

3  qt. 

24. 

19°  31' 
6°  41' 

25. 

18  bu. 

Ipk. 

17  bu. 

3pk. 

26. 

10  A. 

56  sq. 

rd. 

4  A. 

106  sq. 

rd. 

27. 

16  1b. 

4  oz. 

51b. 

12  oz. 

28. 

48  ft. 

3  in. 

27  ft. 

9  in. 

29. 

42'  13" 

35'  5^ 

l^ 

30. 

5  min. 

47  sec. 

2  min. 

.  48  sec. 

31.  Find  the  time  between  Dec.  21,  1620,  and  July  4,  1776. 

32.  How  much  time  elapsed  from  the  beginning  of  the  Civil 
War,  April  14,  1861,  to  the  close  of  the  war,  April  9,  1865  ? 

33.  Washington  was  born  Feb.  22,  1732,  and  died  Dec.  14, 
1799.     How  long  did  he  live  ? 

34.  How  much  time  has  elapsed  from  Oct.  12,  1492,  to  the 
present  time  ? 


SUBTRACTION   OF   COMPOUND  NUMBERS  193 

EXACT  DIFFERENCES  BETWEEN  DATES 

235.    Written 

I.  What  is  the  exact  number  of  days  between  Dec.  16,  1895, 
and  March  12,  1896  ? 

Dec.     15 

Jan        31  There  are  15  days  in  December  after 

■J-,  ■■         oq  the  16th.     January  has  31  days,  February 

29  (leap  year),  and  March  12,  making  87 
March  L£  days.     Always  count  the  last  day. 

87  days.     Ans. 

Note.  —  Every  year  whose  number  is  divisible  by  4  is  a  leap  year,  except  a 
centennial  year,  which  is  a  leap  year  only  when  its  number  is  divisible  by  400  ; 
e.g.  the  year  1896  was  a  leap  year  but  the  year  1900  was  not. 

2.  Mr.  Griffith  bought  a  house,  Feb.  25,  1896,  and  paid  for 
it,  July  12,  1896.  Find  the  exact  number  of  days  between  the 
buying  and  paying  for  the  house. 

3.  Find  the  exact  number  of  days  between  June  25,  1900, 
and  Aug.  24,  1900. 

Mnd  the  exact  time  between : 

4.  Sept.  6,  1896,  and  April  7,  1897. 

5.  Nov.  11,  1898,  and  Dec.  4,  1898, 

6.  Aug.  16,  1907,  and  Dec.  21,  1907. 

7.  July  4,  1896,  and  Aug.  10,  1896. 

8.  Feb.  23,  1897,  and  June  4,  1897. 

9.  Oct.  9,  1899,  and  Feb.  6,  1900. 
10.  Nov.  8,  1905,  and  Oct.  6,  1906. 

II.  A  gardener  planted  an  acre  of  sweet  corn  on  the  tenth 
day  of  May.  It  was  ready  for  market  on  the  third  day  of 
August.     How  many  days  were  required  for  the  corn  to  grow? 


194 


COMPOUND  NUMBERS 


MULTIPLICATION  AND  DIVISION  OF  COMPOUND  NUMBERS 

To  THE  Teacher.  —  Little  time  should  be  spent  upon  multiplication 
and  division  of  compound  numbers.  In  solving  problems  other  than  those 
in  longitude  and  time,  it  is  generally  better  to  reduce  the  compound  num- 
ber to  one  denomination  before  multiplying  or  dividing. 

236.     Written 
1.    Multiply  8  lb.  3  oz.  by  9. 

8  lb.      3  oz  9x3    oz.  =  27  oz.  =  1  lb.  11  oz. 

9  9  X  8    lb.  =  72  lb. 

73  lb.    11  oz.    Product  72  lb.  +  lib.  =  73  lb. 


Multiply : 

2.  7  lb.  9  oz.  by  5.  7. 

3.  15  gal.  1  qt.  by  10.  8. 

4.  14  ft.  11  in.  by  3.  9. 

5.  17  A.  40  sq."  rd.  by  5.  10. 

6.  19  ft.  7  in.  by  4.  ii. 
12.  Divide  32°  15'  30''  by  15. 


1  hr.  20  min.  20  sec.  by  15. 

2  hr.  40  min.  30  sec.  by  15. 

3  hr.    0  min.  30  sec.  by  15. 

4  hr.  19  min.  30  sec.  by  15. 
0  hr.  58  min.  47  sec.  by  15. 


15)32°  16'  30'^ 

2°    9'    6"    Quotient 


82°  4-15    =  2°  and  2°  Remainder. 

2°  =  120'.     120'  +  16'  =  136'. 
136'  -T-  15  =  9'  and  1'  Remainder. 

r=60".     60" +  30"  =  90". 
90"  -4-15=6". 


Divide  by  15  and  test  your  work 

13.  30°  16'  15" 

14.  61°    1'45" 

15.  17°    6'    0" 

16.  2°    2'  15" 

17.  46°  10'    0" 


18.  20°    0'  30" 

19.  20°    5' 45" 

20.  60°  35'  15" 

21.  70°  30'  45" 

22.  100°    1'  30" 


REVIEW  AND  PRACTICE  195 

REVIEW  AND  PRACTICE 
237.      Oral 

1.  At  8)^  a  quart,  what  will  a  bushel  of  berries  cost? 

2.  How  many  oranges  will  50  cents  buy  at  the  rate  of  3  foi 
10  cents? 

3.  f  of  18  is  what  part  of  72?     What  per  cent  of  72? 

4.  From  4|-  bu.  take  3  pk. 

5.  What  will  15  bu.  of  potatoes  cost  if  7|-  bu.  cost  1 6  J? 

6.  At  1 15  a  dozen,  what  will  84  chairs  cost? 

7.  A  cube  5  inches  long  contains  what  part  of  a  cubic  foot  ? 

8.  9  mo.  is  what  part  of  a  year? 

9.  I  sq.  ft.  = sq.  in. 

10.  Bought  a  peck  of  chestnuts  for  80  cents  and  sold  them 
at  5  cents  a  half  pint.  How  much  did  I  gain?  What  per  cent 
of  the  cost  did  I  gain  ? 

11.  25  is  what  part  of  75?     What  per  cent  of  75? 

12.  What  part  of  f  is  |? 

13.  15  gal.  3  qt.  = pt. 

14.  Multiply  7.635  by  10;  by  1000;  by  100. 

15.  Divide  5.8  by  10;  by  100;  by  1000. 

16.  Divide  16.54  by  1654. 

17.  36  lb.  of  coffee  for  i9  is  at  the  rate  of  12  lb.  for  how 
much? 

18.  A  wagon  was  sold  for  $60,  which  was  -J  of  the  cost. 
What  was  the  cost  ?  How  much  was  lost  ?  What  per  cent  of 
the  cost  was  lost? 

19.  How  many  square  rods  are  there  in  -^^  of  an  acre? 

20.  A  ten-acre  lot  is  20  rods  wide.     How  long  is  it? 


196  REVIEW  AND  PRACTICE 

21.  f  of  16  is  I  of  what? 

22.  Andrew  spent  -|  of  his  money  and  had  10  cents  left. 
How  much  had  he  at  first? 

23.  What  is  the  entire  weight  of  three  chickens  if  their  aver- 
age weight  is  2  lb.  11  oz.? 

24.  What  part  of  |^  is  J? 

25.  ^  is  what  part  of  ^? 

26.  Find  20%  of  300;  331%  of  72;  16f  %  of  60;  62J%  of 
16;  90%  of  10. 

27.  Sugar  at  $100  a  ton  is  how  much  a  pound? 

28.  A  house  was  damaged  fSOO  by  fire.     This  was  10%  of 
the  cost.     What  was  the  cost  ? 

29.  87|  %  of  the  cost  of  Glen's  bicycle  was  1 28.     What  was 
the  whole  cost? 

30.  If  9  eggs  cost  27  cents,  what  will  6  doz.  cost  ? 

238.      Written 

1.  The  month  of  January  is  how  many  minutes  long  ? 

2.  If  school  closes  for  the  long  vacation  on  June  24  and 
opens  on  Sept.  6,  how  many  vacation  days  are  there  ? 

3.  If  it  is  now  8.30  a.m.,  what  time  will  it  be  in  20 J  hours  ? 

4.  Find  the  exact  time  from  March  15, 1906,  to  Aug.  2, 1906. 

5.  How  many  weeks,  days,  and  hours  are  there  in  9785  hours  ? 

6.  How  many  rods  of  fence  are  needed  to  inclose  a  square 
field  396  ft.  long  ? 

7.  What  will  it  cost  to  paint  a  ceiling  24  ft.  long  and  16J  ft. 
wide,  at  25  cents  a  square  yard  ? 

8.  What  is  the  length  of  one  side  of  a  square  garden  whose 
perimeter  is  632  feet  ? 


REVIEW  AND   PRACTICE  197 

9.    How  many  square  feet  of  land  are  there  in  a  field  contain- 
ing 21  A.  ? 

10.  What  will  it  cost  at  65  cents  a  square  yard  to  cover  with 
matting  a  floor  21  ft.  long  and  15  ft.  wide  ? 

11.  How  many  feet  of  fence  will  inclose  a  field  46  rd.  by 
31  rd. ? 

12.  Find  the  perimeter  of  a  room  that  is  15'  6"  by  13'  8". 

13.  How  many  rods  are  there  in  2448  inches  ? 

14.  a.  If  steel  rails  are  30  ft.  long,  how  many  are  needed  for 
8  miles  of  street  railroad  track  ?  b.  How  many  tons  do  they 
weigh  if  each  foot  weighs  90  lb.  ? 

15.  At  the  rate  of  660  feet  a  minute,  how  many  miles  will  a 
street  car  run  in  two  hours  ?     (Indicate  and  cancel.) 

16.  What  is  the  cost,  at  $1.80  per  rod,  of  a  fence  inclosing  a 
rectangular  field  50  rd.  by  561  ft.  ? 

17.  Draw  a  line  to  represent  1|  miles,  letting  the  scale  be 
80  rd.  to  an  inch. 

18.  Draw  a  rectangular  building  lot  4  rd.  by  8  rd.,  and  put 
on  the  lot  a  house  44  ft.  by  33  ft.,  and  a  barn  22  ft.  square, 
using  as  a  scale  22  ft.  to  an  inch. 

19.  Three  barrels  of  extract  of  witch  hazel  are  put  up  in 
pint  bottles  and  sold  for  20^  a  bottle.  What  is  received  for 
all  of  it  ? 

20.  A  grocer  bought  40  bbl.  of  cider  at  S  2.50  a  barrel,  made 
it  into  vinegar,  and  sold  it  at  5  ^  a  quart,  a.  How  much  did  he 
gain  ?     b.    What  per  cent  of  the  cost  did  he  gain  ? 

21.  Divide  in  the  shortest  way  the  product  of  36,  40,  144,  8, 
and  160  by  the  product  of  18,  272,  and  6. 

22.  Find  the  least  number  that  will  exactly  contain  27,  60, 
and  24. 


198  REVIEW  AND   PRACTICE 

23.  Find  the  greatest  number  that  will  exactly  divide  231 

and  385. 

24.  In  the  corner  stone  of  a  church  were  cut  these  two  dates, 
MDCCCXXI  — MCMVII.     How  many  years  apart  were  they? 

25.  Mr.  C.  J.  Hogan  has  laid  for  Mr.  Henry  Sumner  a 
piece  of  Portland  cement  walk  50  ft.  long  and  5  ft.  wide  at  12J  ^ 
a  square  foot.     Make  out  the  bill  and  receipt  it. 

26.  Divide:  a.  8.8  by  .0008;  h.  20.005  by  .005. 

27.  Divide  the  sum  of  two  thousand  and  two  thousandths  by 
two  hundredths. 

28.  How  many  times  will  a  bicycle  wheel  rotate  in  going  a 
mile  if  the  wheel  is  7.2  ft.  in  circumference  ? 

29.  The  product  of  two  numbers  is  12.46,  and  one  of  them  is 
24.92.     Find  the  other  number. 

30.  The  product  of  three  numbers  is  .1456.  Two  of  them 
are  2.6  and  .14.     What  is  the  third  ? 

31.  Reduce  to  common  fractions  in  lowest  terms  : 

a.  .025  d.   .06 J  g.  .625  /.   .21 

h.   .0125  e,    .871  h.  .0875  k.  .0495 

c.    .371  /.    .375  i,    .012i  h  4.25 

32.  Express  as  decimals  : 

d.  i^  A.  ff  Z.   14|^         p.    15| 

33.  Find  the  value  of: 

a.  2i  +  32-f.ioff  +  f  of  J  5.     314^x5 


KEVlEW  AKD  PRACTiC:e  l99 

e.  8^x21x^x8  /.  t^xl8| 

34.  A  hardware  merchant  bought  a  bill  of  goods  for  $  560, 
and  marked  them  to  be  sold  at  a  price  which  was  140  %  of  the 
cost.  a.  At  what  price  did  he  mark  the  goods  ?  b.  He  sold 
the  goods  for  90  (fo  of  the  marked  price.  For  what  did  he  sell 
them  ? 

35.  A  jobber  bought  2000  wheelbarrows  at  f  1.25  apiece, 
marked  them  at  115  %  of  their  cost,  and  sold  them  for  95  %  of 
the  marked  price.     What  did  he  receive  for  them  ? 

36.  During  a  storm  1000  bbl.  of  sugar,  or  S^  %  of  the  cargo 
of  a  ship,  were  thrown  overboard.  How  many  barrels  formed 
the  cargo  ? 

37.  Make  and  solve  a  problem  about  $  420  and  35  % . 

38.  Mr.  Fish  paid  city  taxes  to  the  amount  of  f  280,  which 
was  1^  %  of  the  value  of  his  property.  How  much  was  his 
property  worth  ? 

39.  A  load  of  boards  weighing  3500  lb.  were  put  into  a  kiln 
and  dried.  When  taken  out  they  weighed  1^  tons.  What  per 
cent  of  the  weight  of  the  boards  at  first  was  water  ? 

40.  a.  Write  a  problem  in  percentage  in  which  factors  are 
given  to  find  a  product,  h.  Write  another  in  which  one  of  the 
factors  is  to  be  found. 

41.  If  a  pint  of  cream  is  used  in  making  two  gallons  of  ice 
cream,  what  per  cent  of  the  ice  cream  consists  of  other  things 
than  cream  ? 

42.  I  of  my  money  is  $  1260.     What  is  ^  of  my  money  ? 


200 


REVIEW   AN^D  PRACTICE 


43.    This  walk  is  made  of  sawed  Ohio  sandstone, 
cost  at  21^^  per  square  foot. 


Find  its 


7Falk\ 


Walk 


44.  A  man  owned 
f  of  a  mine  and 
sold  I  of  his  in- 
terest for  11710. 
What  was  the  whole 
mine  worth  at  the 
same  rate? 


Walk 


Scale  5  to  ]/i 


45.    A     teamster 

mixed  600  lb.  of 
wheat  bran  with  a 
ton  of  corn  meal 
with  which  to 
feed  his  horses. 
a.  What  per  cent 
of  the  mixture  was  bran  ?     b.   What  per  cent  was  corn  meal  ? 

46.  The  population  of  a  city  to-day  is  130,000,  which  is  125  % 
of  what  it  was  10  years  ago.     What  can  you  find  ?     Find  it. 

47.  Mr.  Hunter's  ranch  in  Wyoming  contains  3200  acres  of 
land,  worth  $25  an  acre. 

a.  How  many  square  miles  of  land  has  he? 

b.  What  is  its  value  ? 

e.  One  fourth  of  the  land  can  be  irrigated  and  made  worth 
$75  an  acre.  Would  it  pay  to  irrigate  the  land  at  a  cost  of 
120,000?     Why? 

48.  a.  Mr.  Hunter  has  6000  sheep.  They  are  worth  $5.25 
apiece  in  the  fall  and  $5.75  apiece  in  the  spring.  What  is  the 
increase  in  the  value  of  the  entire  herd  during  the  winter? 

h.  The  increase  is  what  per  cent  of  the  value  in  the  fall? 
Can  you  tell  why  they  are  worth  more  in  the  spring  ? 


REVIEW  AND  PRACTICE 


201 


The  Ranch 


49.  a.  The  sheep  are  separated  into  two  equal  flocks  and  are 
cared  for  by  two  herders,  two  dogs,  and  a  camp  tender.  Each 
herder  receives  $40  a  month  and  the  camp  tender  850.  The 
supplies  cost  5^  52  per  month.  What  is  the  cost  of  tending  the 
sheep  for  a  year  ?  ! 

h.  On  the  twenty-ninth  day  of  June  the  men  started  with 
their  flocks  for  the  "summer  range"  in  the  mountains  117 
miles  from  the  ranch.  They  traveled 
an  average  distance  of  6J  miles  per 
day.  On  what  day  of  the  year  did 
they  reach  the  summer  range  ? 

e.  In  the  spring  the  sheep  were 
fed  250  tons  of  alfalfa,  worth  |5  a 
ton.  A  ton  of  salt  lasts  them  6 
months  and  costs  I  60.  Add  these 
items  to  the  cost  of  caring  for  the 
sheep,  and  find  the  entire  cost  of 
keeping  these  two  flocks  for  a  year. 

Do  you  know  how  much  a  ton  of  coarse  salt  costs  where 
you  live  ? 

Why  does  it  cost  more  at  this  ranch  ? 


On  the  Move 


202 


REVIEW  AND  PRACTICE 


50.  a.  In  early  summer  5  men  sheared  the  6000  sheep  in  15 
days,  receiving  8  cents  per  fleece  for  their  work.  What  were 
the  average  daily  wages  of  the  shearers? 

h.  The  wool  was  bought  by  a  commission  man  for  a  jobber  in 
the  East  at  22|  cents  per  pound.  The  fleeces  weighed  8  lb. 
apiece,  on  the  average.  What  did  Mr.  Hunter  receive  for  the 
wool  ? 

c.  The  commission  man  charged  the  jobber  10J%  of  this 
sum  for  buying  the  wool.     What  did  the  wool  cost  the  jobber  ? 


A  Wool  Freighter 


d.  The  wool  was  shipped  in  sacks  8  ft.  long  and  4  ft.  wide, 
holding  400  lb.  apiece.  It  was  taken  to  the  railroad  station 
in  loads  of  22  sacks  each,  drawn  by  ten-horse  teams.  How 
many  sacks  were  left  after  3  loads  had  been  drawn  ? 

51.  It  may  be  found  from  the  foregoing  problems  that  Mr. 
Hunter  received  §7426  more  than  he  expended  on  account  of 
his  sheep  business  this  year.  At  the  same  rate,  what  would 
be  a  man's  yearly  profit  from  a  herd  of  48,000  sheep  ? 

52.  If  a  herder  kills  ten  coyotes  in  a  year  and  receives  from 
the  government  a  bounty  of  $3  for  each  one,  how  much  will  he 
add  thereby  to  his  monthly  income? 


ARTICLES  SOLD  BY  THOUSAND  AND  HUNDRED      203 

ARTICLES    SOLD    BY    THE    THOUSAND,    HUNDRED,  OR 
HUNDREDWEIGHT 
239.     Written 

1.  What  is  the  cost  of  8975  bricks  at  1 7  per  M.  ?     (M.  stands 
for  1000.) 

8975  =  8.975  M. 

Since  1  M.  costs  $7,  8.975  M.  cost  8.975  x  $  7,  or  $ ..  Ans. 

2.  What  must  be  paid  for  980  soapstone  pencils  at  f  .30  per 
C.  ?     (C.  stands  for  100.) 

980  =  9.80  C. 

Since  1  C.  costs  $.30,  9.80  C.  will  cost  9.80  x  $.30,  or  | .     Ans. 

3.  Find  the  cost  of  1550  lb.  of  new  buckwheat  flour  at  12.50 
per  cwt. 

1550  lb.  =  15.50  cwt. 

Since  1  cwt.  costs  $  2.50,  15.50  cwt.  cost  15.50  x  $2.50,  or  $ .     Ans. 

Note.  — In  final  results,  a  fraction  of  a  cent,  equal  to  or  greater  than  |  cent, 
is  counted  a  whole  cent.    A  fraction  which  is  less  than  ^  cent  is  dropped. 

4.  Find  the  cost  of  each  of  the  following  items  and  the  total  cost 
of  all  of  them  : 

a.  27,325  bricks  at  f  5.15  per  M. 

b.  4900  cu.  ft.  of  gas  at  $1.20  per  M. 

c.  583  lb.  sugar  at  14.75  per  C. 

d.  4900  tomato  plants  at  $1.50  per  M. 

e.  1000  laths  at  40  ^  per  C. 

/.  3125  cu.  ft.  of  city  water  at  $.14  per  C. 

g.  1  T.  fiber  paper  at  $2.50  per  C. 

h.  5600  paper  bags  at  $2.90  per  M. 

i.  16  boxes  of  envelopes,  250  in  a  box,  at  $1  per  M. 


204 


MEASUREMENTS 


y.   850  Japanese  napkins  at  i.l8  per  C. 

k.    10  bbl.  American  lump  salsoda,  375  lb.  in  a  barrel,  at 
1.80  per  cwt. 

I.    6500  No.  3  butter  trays  at  11.60  per  M. 

m.  8950  7-inch  picnic  plates  at  $1.75  per  M. 

n.   13,500  cedar  shingles  at  |4.10  per  M. 

0,    675  lb.  light  manila  bread  paper  at  13.75  per  cwt. 


MEASUREMENTS        » 
AREAS  OF  PARALLELOGRAMS 
'240.    A  plane  figure  hounded  hy  four  straight  lines  is  a  quadri- 


lateral;  e.g. 


241.     Lines  that  are  the  same  distance  apart  throughout  their 
whole  length  are  parallel  lines ;  e.g. 


242.  A  quadrilateral  whose  opposite  sides  are  parallel  is  a 
parallelogram.    Which  of  the  above  figures  are  parallelograms  ? 

243.  A  parallelogram  that  has  four  right  angles  is  a  rectangle. 
Which  of  the  above  figures  are  rectangles  ? 


244.    Two  lines  that  meet  to  form  a  right  angle  are 
perpendicular  to  each  other. 


245.     The  side  on  which  a  figure  is  supposed  to  rest  is  its  base. 


AREAS  OF  PARALLELOGRAMS 


205 


246.      The  perpendicular  distance  from  the  highest  point  of  a 
figure  to  the  base,  or  to  the  base  extended,  is  its  altitude ;  e.g. 


'^ 


247.  Figures  are  read  by  means  of  letters  placed  at  their  angles. 
Thus,  Fig.  1  is  read,  ''Oblong  ABOD.''  Fig.  2  is  read,  "Tri- 
angle ABO.''  Read  Figs.  3  and  4.  The  base  of  the  triangle  in 
Fig.  2  is  AC.     The  altitude  of  Fig.  1  is  DQ  ov  AB, 

248.  Oral 

1.  In  Fig.  5  the  part  ^compares  how  with  the  part  Jf  ? 

2.  The  area  of  the  par- 
allelogram ABCB  com- 
pares how  with  the  area  of 
the  parallelogram  EFOD  ? 

3.  What  is  the  base  of 
each  of  these  parallelo- 
grams ? 

4.  What  is  the  altitude  ?     What  is  the  area  ? 

5.  Cut  from  paper  different  shaped  parallelograms. 

6.  Change  them  to  rectangular  parallelograms  by  cutting  off 
a  part  similar  to  ^and  placing  it  like  M. 

7.  How  is  the  area  of  a  rectangle  found  ? 

8.  If  the  base  of  a  rectangle  is  its  length,  the  altitude  is  what  ? 


A                E 

B 

F 

\k  1, 

\ 

V  M    i 

\     !< 

\     1' 

\        I'* 

N' 

Fig.  5 


206 


AREAS  OF  PARALLELOGRAMS 


9.    If  we  know  the  base  and  altitude  of  a  rectangle,  how 
may  we  find  the  area  ? 

10.  Since  any  parallelogram  may  be  made  into  a  rectangle  of 
the  same  base  and  altitude,  how  may  we  find  the  area  of  a  par- 
allelogram ? 

249.  The  area  of  a  parallelogram  is  equal  to  the  product  of  its 
base  and  altitude. 

250.  Oral 

The  following  examples  relate  to  parallelograms. 

Fill  in  the  missing  number : 
Base  Alt.  Area  Base  Alt.  Area 

1.  8  ft.        6  ft.         5. 3J  in.      7  sq.  in. 

2.  71  in.     8  in.        6.    2  ft. 

3.  5  mi.      4  sq.  mi.  7.    3|  ft. 

4.    6  yd.       96  sq.  yd.       8.    1  yd.       15  sq.  ft. 


Alt. 

3J  in. 
6  in. 

8  ft. 

1yd. 


251.    The  following  parallelograms  are  drawn  to  the  scale  of 
1'  to  ^".     Find  their  areas  in  square  feet. 


AREAS  OF  TRIANGLES 


207 


AREAS  OF   TRIANGLES 


252.    A    'plane  figure    hounded    hy  three   straight    lines   is   a 
triangle ;  e.g. 


253.    Oral 


Fig.  1 


Fig.  2 


Fig.  3 


Fig.  4 


1.  Figures  1,  2,  3,  and  4  are  what  kind  of  figures  ?     What 
-kind  of  figures  are  A  and  B  2 

2.  In  each  of  the  above  figures,  how  does  A  compare  with  B? 

3.  In  each  of  the  above  figures,  how  do  the  base  of  the 
triangle  and  the  base  of  the  parallelogram  compare  ? 

How  do  the  altitude  of  the  parallelogram  and  of  the  triangle 
compare  ? 

4.  Fold  a  paper  once.  By  cutting  through  both  leaves  of 
the  paper  at  the  same  time,  you  may  cut  out  two  triangles. 
How  do  they  compare  ?  Put  them  together  so  as  to  make  a 
parallelogram. 

5.  Repeat  this  process,  cutting  triangles,  each  pair  of  which 
differ  in  shape  from  the  other  pairs. 

Each  triangle  is  what  part  of  the  parallelogram  having  the 
same  base  and  altitude? 

6.  How  is  the  area  of  the  parallelogram  found?  Of  the 
triangle?  ^ 


208 


AREAS   OF  TRIANGLES 


The  area  of  a  triangle  is  equal  to  one-half  the  product  of  its 
base  and  altitude. 

7.  The  following  triangles  are  drawn  to  a  scale.  The  bases 
and  altitudes  of  the  triangles  which  they  represent  are  indicated 
in  the  drawings.     Find  their  areas. 

17 


nrd. 


254.      Written 

1.    Find  the  areas  of  triangles  having  the  following  dimen- 


sions,  using  cancellation  where 

possible.     When  the  base  and 

altitude  are    of   different    denominations,  make 

them  similar 

before  multiplying. 

Base 

Alt. 

Basb 

Alt. 

a.    51  ft. 

42  ft. 

y.    15  yd. 

15  in. 

b,    27  in. 

15  in. 

h.    2  mi. 

160  rd. 

c.    6fft. 

7Jft. 

L    11  mi. 

1201  rd. 

d.    3  ft.  7  in. 

2  ft.  9  in. 

m.   4  rd. 

3J  rd. 

e.    9fin. 

16  in. 

71.    2  yd. 

50  in. 

/.   18  yd. 

27  in. 

0.    3^rd. 

12  ft. 

g,    12  ft. 

Hyd. 

p,    101  yd. 

5  ft.  4  in. 

h,    14  rd. 

7  yd. 

q.    7  yd. 

7  ft. 

i.    1  mi. 

80  rd. 

r.    17fin. 

A^d. 

MEASUREMENTS:   THE   CORD 


209 


2.    The  above  figures  are  drawn  to  the  scale  of  10  ft.  to  J  in. 
Find  their  bases,  altitudes,  and  areas. 


THE  CORD 

255.     The  pile  of  wood  in  the  center  of  this  picture  is  8  ft. 
long,  4  ft.  wide,  and  4  ft.  high.     How  many  cubic  feet  does  it 


contain  ? 


128  cubic  feet  =  1  cord. 


In  the  forest,  fuel  wood  for  market  is  generally  cut  in  4-foot 
lengths  like  that  in  the  picture,  so  that  a  pile  4  ft.  high  and 


210  MEASUREMENTS:   THE  CORD 

8  ft.  long  contains  a  cord  of  128  cu.  ft.  The  term  cord^  how- 
ever, is  often  used  to  mean  any  pile  of  wood  that  is  8  ft.  long 
and  4  ft.  high,  whatever  may  be  the  length  of  the  sticks. 

256.  Oral 

1.  Hold  your  hand  above  the  floor  high  enough  to  show  the 
height  of  the  pile  of  wood  in  the  picture.  Stand  as  many  feet 
from  the  side  of  the  room  as  the  pile  is  long.  Show  with  your 
hands  the  length  of  the  sticks. 

2.  How  many  cubic  feet  are  there  in  a  pile  of  wood  8  ft. 
long  and  4  ft.  high,  if  the  sticks  are  1  ft.  long  ?  Show  with 
your  hands  the  height  of  this  pile.  Show  its  width.  Walk  far 
enough  to  show  its  length. 

3.  How  many  cubic  feet  would  there  be  in  the  above  pile  if 
the  sticks  were  2  ft.  long  ?  Show  the  width  of  the  pile  with 
your  hands. 

4.  How  many  cubic  feet  would  there  be  in  the  pile  if  the 
sticks  were  3  ft.  long  ?  1^  ft.  long  ?  Show  these  lengths  with 
your  hands. 

257.  Written 

1.  Using  cancellation,  find  the  number  of  cords  in  a  pile  ot 
4-foot  wood : 

a.  20  ft.  long  and  8  ft.  high. 
h.     64  ft.  long  and  4  ft.  high. 

c.  12  ft.  long  and  6  ft.  high. 

d.  100  ft.  long  and  7  ft.  high. 

e.  26  ft.  long  and  5  ft.  high. 
/.  40  ft.  long  and  4  ft.  high. 
g.  72  ft.  long  and  7  ft.  high. 
h.  18  ft.  long  and  2  ft.  high. 


MEASUREMENTS:   BUILDING  WALLS 


211 


2.  If  a  pile  of  4-foot  wood  is  48  ft.  long,  how  high  must  it  be 
to  contain  9  cd.  ? 

3.  What  must  be  paid  for  enough  4-foot  wood  to  fill  a  shed 
26  ft.  long,  16  ft.  wide,  and  12  ft.  high  at  14.50  a  cord  ? 

4.  In  the  yard  of  a  certain  tannery  there  is  a  pile  of  bark 
100  ft.  long,  24  ft.  wide,  and  10  ft.  high.  How  many  cords  of 
bark  are  there  in  the  pile  ? 


BUILDING  WALLS 

258.  There  are  no  universal  rules  for  the  measurement  of 
masonry.  Some  masons  measure  around  the  outside  of  a  cellar 
wall  to  determine  its  dimensions,  while  others  make  allowance 
for  the  corners.  The  method  of  measurement  should  be  specified 
in  the  contract  in  every  case. 

Quantities  of  uncut  stone  are  bought  by  the  cord,  and  usually 
99  cu.  ft.  are  taken  for  a  cord. 

From  21  to  23  bricks  8^'  X  4'^  x  2''  are  estimated  to  make  a 
cubic  foot  of  brick  wall. 

259.  Written 


7' 6" 

\_1 

\\  \ \ \ \X vX'xx ^\< v<:\<;\^-^^ff'«' 

\\\ 

1 

1     {     1     1     1     1     1     I     1     1 

\\X\ 

\V 

x\ 

s 

1.    a.  How  many  cubic  feet  does  this  wall  contain  ? 
h.    If  21  bricks  make  a  cubic  foot  of  wall,  how  many  bricks 
are  used  in  this  wall  ? 


212  MEASUREMENTS:   FLOOR   COVERING 

c.  What  do  they  cost  at  1 6.30  per  M.? 

d.  At  the  same  rate,  what  would  be  the  cost  of  brick  for  a 
partition  wall  50  ft.  long,  11  ft.  high,  and  12  in.  thick  ? 

e.  When  22  bricks  2'^  x  4''  x  8''  make  a  cubic  foot  of  wall, 
how  many  cubic  inches  of  mortar  are  used  ? 

2.  A  house  built  in  the  form  of  a  rectangle  36  ft.  by  21  ft. 
has  a  cellar  8|-  ft.  deep.     The  cellar  wall  is  1  ft.  6  in.  thick. 

a.    Draw  a  plan  of  the  cellar  wall. 

h.  Find  the  number  of  cubic  feet  to  be  paid  for  in  the  cellar 
wall,  measuring  around  the  outside,  and  making  no  allowance 
for  the  corners. 

e.  If  the  cellar  wall  extends  3  ft.  above  the  grolind,  how 
many  cubic  yards  of  earth  were  removed  to  make  the  cellar  ? 

d.  What  was  the  cost  of  the  brick  at  $6.50  per  M.  for  a 
partition  wall  12  in.  thick,  from  end  to  end  through  the  middle 
of  the  cellar,  allowing  22  bricks  for  a  cubic  foot  ? 

e.  What  was  the  cost  of  a  Portland  cement  floor  in  one  half 
of  this  cellar  at  10  ^  per  square  foot  ? 

3.  Find  the  cost  of  the  stone  for  a  wall  297  ft.  long,  8  ft. 
high,  and  IJ  ft.  thick  at  16.50  a  cord  (99  cu.  ft.). 

FLOOR  COVERING 

260.  The  exact  number  of  yards  of  carpet,  matting,  or  other 
covering  to  be  purchased  for  any  given  floor  is  diflicult  to  deter- 
mine  because  of  the  waste  in  fitting  and  in  matching  figures. 
We  may  obtain  a  fairly  correct  estimate  by  dividing  the 
number  of  square  yards  of  surface  to  be  covered,  by  the 
width  of  the  strips,  in  yards,  and  then  adding  a  certain 
amount  for  waste. 

A  yard  of  carpet,  matting,  etc.,  is  a  yard  of  the  length  of 
the  roll,  regardless  of  its  width. 


MEASUREMENTS:    FLOOR  COVERING 


213 


261.    Oral 

1.    A  piece  of  carpet  1  yd.  long  and  |  yd.  wide  will  cover  how 

Qyd. 


much  surface?     Draw  it  full  size  on 
the  blackboard. 

2.  What  is  the  area  of  this  floor  ? 

3.  If  a  yard  of  carpet  |  yd.  wide 
will  cover  |  of  a  square  yard  of  floor, 
how   many   yards   of   such   carpet  will    cover    18   sq.   yd.    of 
floor  ? 

4.  How  many  yards  of  matting  1  yd.  wide  will  cover  the 
same  floor  ? 

262.    Written 

1.    If  this  floor  is  covered  with  carpet  |  yd.  wide,  how  many 
strips,   running  lengthwise,   must  be 
purchased  ? 

Note.  —  When  a  part  of  the  width  of  a 
strip  is  needed,  a  whple  strip  must  generally 
be  purchased. 


2.  How  many  yards  of  carpet  must 
be  purchased  for  this  floor  ? 

3.  Draw  a  diagram  of  each  of  the  floors  whose  dimensions  are 
given  below,  and  compute  the  number  of  yards  of  material  to  be 

i       purchased  to  cover  it,  running  the  strips  the  longer  way  of  the  floor : 


Dimensions 

Width  of  Material 

a. 

81  yd.  X  5  yd. 

1yd. 

b. 

15  ft.  X  3  yd. 

2  ft.  3  in. 

c. 

101  ft.  X  18  ft. 

fyd. 

d. 

18  ft.  X  24  ft.  6  in. 

1yd. 

e. 

17  ft.  X  27  ft.        . 

36  in. 

214  MEASUREMENTS:   FLOOR  COVERING 


/. 

9  ft.  X  28  ft. 

|yd. 

9' 

13  ft.  3  in.  X  15  ft. 

fyd. 

h. 

12  ft.  X  16  ft. 

1yd. 

i. 

19^  ft.  X  29  ft. 

fyd. 

h 

15  ft.  9  in.  X  19  ft. 

2  ft.  3  in. 

k. 

11  ft.  3  in.  X  14  ft. 

27  in. 

I. 

20  ft.  X  10  yd. 

1yd. 

m. 

16  yd.  X  6  yd. 

54  in. 

n. 

20  ft.  X  38  ft. 

fyd. 

0. 

16  ft.  X  22  ft. 

1yd. 

V' 

15  ft.  X  18  ft.  3  in. 

27  in. 

<!' 

29  ft.  X  16  ft.  6  in. 

fyd. 

r. 

14  ft.  X  20  ft. 

iyd. 

s. 

13  ft.  8  in.  X  19  ft.  6  in. 

1yd. 

t. 

6  yd.  X  23  ft. 

ijyd. 

u. 

100  ft.  X  75  ft. 

1  ft.  8  in. 

V. 

31  yd.  2  in.  x  13  yd.  1  ft.  6  in. 

fyd. 

4.  What  is  the  expense  of  covering  a  kitchen  floor  12  ft.  x 
13|  ft.  with  inlaid  linoleum  at  $1.40  per  square  yard,  allowing 
11  sq.  yd.  for  waste  in  matching  the  pattern  ? 

5.  Find  the  cost  of  covering  a  porch  floor  9  ft.  by  30  ft.  with 
plain  cocoa  matting  54  in.  wide  at  50^  a  square  yard. 

6.  The  living  room  of  a  summer  cottage  is  17  ft.  x  24  ft., 
and  the  floor  is  covered  with  plain  grass  matting  1  yd.  wide, 
laid  so  as  to  make  no  waste.  (Which  way  must  the  strips  run?) 
Find  the  cost  at  40/  a  yard. 

7.  Harold's  bedroom  is  lOJ  ft.  x  13|^  ft.  It  is  covered  with 
matting  1  yd.  wide,  costing  45  /  a  yard,  pieced  so  as  to  make 
no  waste  except  in  turning  under  at  the  ends.     Carpet  paper, 


MEASUREMENTS;  FLOOR  COVERING 


215 


costing  5/  a  square  yard,  is  laid  under  the  matting.  Two 
5-yard  pieces  of  braid,  costing  2^  a  yard,  are  used.  Harold 
does  the  work,  wasting  \  yd.  of 
matting  and  -^  sq.  yd.  of  paper. 

a.  What  is  the  entire  cost  ? 

b.  Draw  a  diagram  of  the  room 
showing  how  the  matting  is  laid 
and  pieced. 

8.  This  rug  is  made  of  Royal 
Wilton  carpet  costing  $2.60  a 
yard.  The  making  and  sizing  cost 
121  f^  a  yard.     22  yd.  of  carpet 

braid  were  used  at  a  cost  of  5^  a  yard,  sewed  on.  Find  the 
entire  cost  of  the  rug,  allowing  |  yd.  for  waste  in  matching 
the  pattern. 


19'9" 

i                          16' 

r 

l_    _ 

This  is  Lucy's  room  drawn  to  the 
scale  of  I"  =  V. 


9.  a.  What  would  a 
hard-wood  floor  in  this 
room  cost  at  16J^  per 
square  foot? 

b.  Find  the  cost  of 
carpeting  this  room  with 
three-ply  ingrain  carpet 
1yd.  wide  at  90  (^  per  yard, 
paying  5^  per  yard  for 
making  and  laying,  and 
5^  per  square  yard  for 
carpet  paper. 

e.  Find  the  cost  of 
cleaning  the  carpet  at 
8^  per  square  yard. 


216 


MEASUREMENTS:  PLASTERING 


PLASTERING 

263.  There  is  among  builders  no  universal  rule  for  comput. 
ing  the  amount  of  plastering  in  the  walls  and  ceilings  of  a 
building.  Some  contractors  deduct  the  entire  surface  of  open- 
ings, such  as  doors,  windows,  etc.,  and  some  only  one  half  of 
such  surface. 


264>.    Written 

1.  The  scale  of  this  drawing  is  -^q  '^  = 
1'.  Make  a  drawing  like  this,  using  ^" 
for  1'.  Cut  it  out,  and  fold  it  on  the 
dotted  lines,  making  the  model  of  a  room. 


12' 
End 
Wall 


16' 
Side 
Walt 


12' 

End 
Walt 


16' 
Ceiling 


16' 
Side 
Wall 


a.  What  is  the  entire  length  of  the  end  and  side  walls  ? 
What  is  the  hei^  ht  ? 

b.  How  many  square  feet  are  there  in  all  the  walls  ? 

c.  How  many  square  feet  are  there  in  the  ceiling  ? 

d.  How  many  square  feet  are  there  in  the  walls  and  ceiling 
together  ? 

e.  How  many  square  yards  are  there  in  all  ? 

/.  What  will  it  cost  to  lath  and  plaster  this  room  at  85  cents 
a  square  yard,  taking  out  5 J  square  yards  for  openings  ? 

2.  a.  What  will  it  cost  to  lath  and  plaster  a  room  15'  by  18' 
and  9'  high  at  f  .30  a  square  yard,  allowing  10  sq.  yd.  for  doors 
and  windows?  h.  What  will  it  cost  at  |.28  a  square  yard, 
making  no  allowances  ? 


MEASUREMENTS:   WALL  COVERINGS  217 

3.  Measure  your  schoolroom  to  the  nearest  half,  third,  or 
fourth  of  a  foot,  and  estimate  the  cost  of  lathing  and  plastering 
it  at  40)^  a  square  yard.  Allow  3  sq.  yd.  for  each  door  and 
window. 

4.  Estimate  the  cost  of  plastering  other  rooms  in  your  school 
building  in  the  same  way. 

5-6.  Measure  two  rooms  in  your  house.  Find  the  cost  of 
plastering  the  walls  at  30  cents  a  square  yard,  allowing  2  sq.  yd. 
for  each  door  and  each  window.     Bring  your  work  to  school. 

7.  Find  the  cost  of  lathing  and  plastering  the  walls  and  ceil- 
ings of  a  room  27  ft.  by  18  ft.  and  9|  ft.  high,  allowing  for 
4  windows,  each  3  ft.  by  6  ft,  and  3  doors,  each  3  ft.  by  8  ft., 
at  32  cents  per  square  yard. 

WALL   COVERINGS 

265.  A  roll  of  figured  wall  paper  is  usually  8  yards  long  and 
J  yard  wide.    How  many  square  yards  of  paper  does  it  contain  ? 

Ingrain  paper  is  30  inches  wide. 

Paper  hangers  generally  estimate  that  a  roll  of  paper  will 
cover  from  30  to  34  square  feet,  after  allowing  for  waste. 
Woven  wall  coverings  are  sold  by  the  square  yard. 

266.  Written 

1.  Fanny's  mother  wished  to  decorate  a  room  in  her  house, 
and  Fanny  estimated  the  cost.  The  dimensions  were  16  ft.  by 
21  ft.,  and  9  ft.  9  in.  high.  There  were  2  doors  and  3  windows, 
each  estimated  at  2  sq.  yd. 

a.  The  paper  was  estimated  to  cover  30  sq.  ft.  per  roll. 
How  many  rolls  were  needed  for  the  walls  ?  (You  cannot  buy 
a  part  of  a  roll.) 

b.  What  would  the  paper  cost  at  25^  a  roU  ? 


218  MEASUREMENTS:   WALL  COVERINGS 

c.  The  molding  to  extend  all  around  the  room  at  the  top  of 
the  walls  was  sold  only  in  12-foot  lengths,  and  cost  4J^  a 
foot.     What  would  it  cost  ? 

c?.  In  preparation  for  tinting,  the  ceiling  was  to  be  lined 
with  paper  at  10^  a  roll  of  30  sq.  ft.  What  would  this  paper 
cost? 

e.  They  expected  to  use  two  packages  of  tinting  material 
costing  30^  a  package;  the  putty,  glue,  flour,  etc.,  were  esti- 
mated  at  Q^^\  and  the  labor  at  two  days'  work  for  two  men 
at  $3.50  per  day  for  each  man.  What  should  have  been  the 
total  of  Fanny's  estimate  ? 

/.  If  they  decide  to  add  to  the  ceiling  some  relief  work 
which  costs  f  3.81,  and  the  men  can  put  it  on  in  ^  of  a  day,  how 
much  must  Fanny  add  to  her  estimate  ? 

2.  Before  being  tinted,  a  ceiling  12'  x  17'  was  covered  with 
sheeting  2  yd.  wide.  What  did  the  sheeting  cost  at  25  ^  a 
lineal  yard  ? 

3.  a.  How  much  money  is  needed  to  buy,  at  40^  a  square 
3^ard,  enough  crash  to  cover  the  four  walls  of  a  room  18'  x  32' 
and  10  ft.  high,  allowing  190  sq.  ft.  for  baseboard  and  open- 
ings, and  not  purchasing  a  fraction  of  a  square  yard  ? 

h.  What  would  it  cost  to  paint  the  walls  and  ceiling  of  this 
room  at  f  .23  a  square  yard  ? 

4.  Estimate  the  cost  of  painting  the  walls  and  ceiling  of 
your  schoolroom  at  f  .25  a  square  yard. 

5.  Select  a  room  in  your  own  home;  measure  it.  Find  the 
cost  of  decorating  it  as  your  mother  would  like  to  have  it  done. 
Ask  her  to  tell  you  what  she  would  like  to  have  put  on  the 
walls ;  then  you  make  the  measurements,  compute  the  amount 
of  material  and  labor,  and  the  cost. 

Make  and  solve  other  problems  in  papering. 


LUMBER  MEASURE  219 

LUMBER  MEASURE 

267.  A  piece  of  wood  1  ft.  long,  1  ft.  wide,  and  1  in.  thick 
is  a  board  foot  (bd.  ft.). 

To  THE  Teacher.  —  As  material  for  this  lesson,  a  real  board  foot  —  a 
piece  of  board  exactly  1  ft.  long,  1  ft.  wide,  and  1  in.  thick  —  should  be  pro- 
vided. Refer  to  it  in  obtaining  answers  to  the  oral  questions  below  and 
whenever  pupils  seem  to  answer  wide  of  the  mark  in  this  subject.  This  is 
very  important. 

268.  Oral 

1.  How  many  inches  long  is  a  board  foot  ?  How  many- 
inches  wide  ?  How  many  cubic  inches  does  a  board  foot 
contain  ? 

2.  How  many  board  feet  piled  one  upon  another  would  make 
a  cubic  foot  ?  Show  with  your  hands  how  wide,  long,  and  high 
this  pile  would  be. 

3.  A  board  1  in.  thick,  1  ft.  wide,  and  6  ft.  long  contains 
how  many  board  feet  ?  Draw  it  full  size  on  the  blackboard, 
and  mark  off  the  board  feet. 

4.  If  the  board  in  question  3  were  twice  as  thick,  how  many 
board  feet  would  it  contain  ?  How  many  inches  thick  would 
it  be  ? 

5o  If  it  were  five  times  as  thick,  how  many  board  feet  would 
it  contain? 

6.  How  many  board  feet  are  there  in  a  piece  of  board  1  ftc 
wide,  16  ft.  long,  and  1  in.  thick? 

7.  If  this  piece  of  lumber  were  2  in.  thick,  how  many  board 
feet  would  it  contain  ?  3  in.  ?  4  in.  ?  5  in.  ?  6  in.  ? 

8.  A  piece  of  inch  board  3  ft.  long  must  be  how  wide  to  con- 
tain 1  bd.  ftt 


220 


LUMBER  MEASURE 


9.  A  cubic  foot  of  wood  could  be  sawed  into  how  many 
board  feet  if  there  were  no  waste  in  sawing?  The  number  of 
board  feet  in  any  piece  of  lumber  is  how  many  times  as  great 
as  the  number  of  cubic  feet  ? 

269.  We  may  find  the  number  of  board  feet  in  a  piece  of 
lumber  by  multiplying  the  number  of  cubic  feet  by  12.  The 
rule  commonly  used  by  dealers  and  mechanics  gives  the  same 
result,  and  is  stated  as  follows  : 

To  find  the  number  of  hoard  feet  in  any  piece  of  lumber^  mul- 
tiply together  its  three  dimensions,  two  of  them  expressed  in  feet 
and  the  other  in  inches. 

Lumber  that  is  less  than  1  in.  thick  is  counted  as  1  in.  thick 
in  measuring. 

270.  Oral 

1.  About  how  many  feet  of  lumber  (board  feet)  are  there  in 
the  top  of  your  desk  ?  The  teacher's  desk  ?  One  end  of  the 
bookcase  ?    The  cupboard  door  ?    All 

the  shelves  in  the  bookcase  ? 

2.  How  much  lumber  is  used  in 
making  this  box  with  cover  ?  (Take 
outside  measurements.) 

3.  Estimate  the  amount  of  lum- 
ber in  a  cubical  box,  including  the 
cover,  made  of  |^-inch  lumber,  the 
length  of  the  box  being  three  feet. 

4.  The  door  in  this  hayloft  is 
4|'x4'.  Two  battens  across  the 
inside  are  4'  x  6'^  How  many  feet 
of  lumber  are  used  in  the  door,  the 
lumber  being  1''  thick  ? 


LUMBER  MEASURE 


221 


271.     Written 

1.  How  many  board  feet  are  there  in  a  piece  of  timber 
18' X  le^'x  3^'?     (18'xi|'x  3'^= bd.  ft.) 

2.  The  floor  of  a  tent  12  ft.  by  16  ft.  is  made  of  boards 
1  in.  thick  laid  close  together,  a.  How  many  feet  of  lumber 
are  used  ?  h.  How  much  is  it  worth  at  $  30  per  M.  (thousand 
feet)  ? 


a.  Find  the  number  of  feet  of  lumber  in  this  stick.  Walk 
as  far  on  the  floor  as  the  length  of  this  piece  of  timber. 
Show  with  your  hands  how  high  it  is.  Show  how  wide  it  is. 
How  many  feet  of  lumber  does  it  contain  ? 

h.    What  are  twenty  such  sticks  worth  at  f  26  per  M.  ? 


a.  Find  the  amount  of  lumber  in  this  plank.  5.  Find  the 
cost  of  ten  such  planks  at  |  30  per  M. 

5.  The  floor  of  your  schoolroom  is  |^  in.  thick,  a.  If  no 
allowance  is  made  for  sawing  and  matching,  how  many  feet  of 
lumber  are  there  in  the  floor?  h.  If  \  of  the  lumber  was 
wasted  in  sawing  and  matching,  the  floor  contains  only  ^  of  the 
lumber  that  was  bought.  How  much  lumber  was  bought  ? 
c.    What  did  it  cost  at  ^  42  per 


M.? 

6.  A  fence  like  this,  6  ft.  high, 
extends  around  two  sides  and  one 
end  of  a  rectangular  garden  40 

ft.  by  t)^  ft.     Draw  a  diagram  of  the  garden,     a.  How  many  feet 
of  boards  were  used  ?    h.  How  much  did  they  cost  at  $  27  per  M.  ? 


222 


LUMBER  MEASURE 


7.    The  floor  of  this  bridge  is  14  ft.  by  8  ft.,  and  made  of  oak 

planks    3    in.    thick.      What 
did  they  cost  at  1 45  per  M.  ? 

8.    Is  there  a  board  fence 
at  the  rear  or  side  of  your 
school  ground  ?     If  so,  find 
the  number  of  feet  of  boards 
in  it,  as  a  part  of  to-morrow's  lesson. 

9.    Find  the  cost  of  each  of  the  following  quantities  of  lumber  : 


Number  of  Pieces 

Dimensions 

Price  per  M 

a. 

21 

3^^x 

12''  X  18' 

124 

h. 

10 

^"  X 

6"  X  20'    . 

$26 

c. 

75 

2"x 

4"  X  20' 

$26 

d. 

6 

10^' X 

14"  X  30' 

$  32" 

e. 

3 

11'' X 

11"  X  10' 

$35 

/. 

49 

l''x 

5"  X  16' 

$30 

9- 

60 

2''x 

10"  X  14' 

$24 

h. 

72 

2''x 

8"  X  14' 

$24 

i. 

m 

2''x 

6"  X  14' 

$25 

h 

121 

rx 

3"  X  12' 

$28 

h. 

2 

6''x 

10"  X  22' 

$25 

L    . 

4 

6"x 

10"  X  16' 

$25 

m. 

4 

2''x 

8"  X  18' 

$26 

n. 

8 

S^x 

8"  X  14' 

$27 

0. 

16 

y'x 

10"  X  12' 

$40 

P- 

7 

2J''x 

8"  X  16' 

$45 

?• 

600 

rx 

51"  X  10' 

$42 

REVIEW   AND  PRACTICE  223 

REVIEW  AND  PRACTICE 
272.      Oral 

1.  Read:  1,003,050 ;  XCIX;  MCM ;  CDIV;  CLI ;  2,050,205. 

2.  What  is  the  greatest  common  divisor  of  12,  28,  and  56? 

3.  What  is  the  smallest  number  that  exactly  contains  4,  5, 
and  8  ? 

4.  Give  five  multiples  of  11. 

5.  Name  all  the  composite  numbers  smaller  than  30. 

6.  Name  the  prime  factors  of  42. 

7.  What  common   fraction  in  lowest  terms   equals    .33J? 
.871?  .625?  .37-1?  .20?  .80?  .60? 

8.  What  decimal  is  equal  to  f  ?  J  ?  |  ?  f  ? 

9.  Give  results:    .8x7;     .12-J-6;      1.5x.6;      1.03x5; 
I       7.01  X. 6. 

10.  Give  results  :    2000  ^  20  ;       3.528  x  100  ;       53  -r- 1000 ; 
950-50;  2.3x1000. 

11.  10  is  how  many  times  ^  ? 

12.  What  is  the  cost  of  24  oranges,  when  8  oranges  cost  25 
cents  ? 

13.  What  is  the  difference  between^  of  10  and  ^  oi  10? 

14.  If  21  yd.  of  cloth  cost  1 7 J,  what  will  5  yd.  cost  ? 

15.  f  of  20  are  how  many  times  4  ? 

16.  J  is  contained  in  7  how  many  times? 

17.  Give  results:   J-i;  ^-^1   1-i;  -i  +  J;  l  +  l;  |  +  1; 
A  X  i' 

18.  What  is  the  cost  of  three  bananas  at  20  cents  a  dozen  ? 


224  REVIEW  AND  PRACTICE 

19.  What  will  4  doz.  steel  screws  cost  at  15  i  a  gross  ? 

20.  If  3  boys  can  shovel  a  walk  in  15  minutes,  how  long 
should  it  take  1  boy  ? 

21.  When  3  eggs  cost  5  cents,  what  is  the  cost  per  dozen  ? 

22.  A  rectangle  18  in.  by  2  in.  contains  what  fraction  of  a 
square  foot  of  surface? 

23.  \  of  25  is  what  part  of  30? 

24.  Give  results  :  41  -  f ;  |  of  f ;  4  -  If 

25.  A  man  had  a  sum  of  money.     He  earned  \  as  much  and 
then  had  |9.     How  much  had  he  at  first  ? 

26.  \  inch  is  what  part  of  a  foot  ? 

27.  40  rd.  are  what  part  of  a  mile  ? 

28.  40  sq.  rd.  are  what  part  of  an  acre  ? 

29.  How  many  boys  are  there  in  a  class  of  45  pupils  if  \  of 
the  pupils  are  girls  ? 

30.  How  many  cubic  feet  of  stone  are  there  in  a  stone  wall 
8'  X  4'  X  2'  ? 

31.  Ethel  cut  20  roses  one  morning,  of  which  40  %  were  red. 
How  many  red  roses  did  she  cut? 

32.  One  dozen  is  what  per  cent  of  one  gross  ? 

33.  A  peddler  sold  36  pencils  at  the  rate  of  2  for  5  cents. 
What  did  he  receive  ? 

34.  48^3  +  36-50  =  ? 

35.  28  days  are  called  a  lunar  month.     One  week  is  what 
per  cent  of  a  lunar  month  ? 

36.  How  many  days  were  there  in  February,  1906  ?    In  Feb- 
ruary, 1493  ? 


REVIEW  AND  PRACTICE  225 

273.    Written 

1.  See  how  quickly  you  can  obtain  correct  answers  and  test 
them: 

a.  The  addends  are  4,798,  6,430,  895,  49,785,  231,  8,942, 
700,  98,346,  209,  98,020,  49,  816.     Find  the  sum. 

h.  The  product  is  910,386,  one  factor  is  4,398.  Find  the 
other  factor. 

e.    7,695  and  493  are  the  factors  that  make  what  number  ? 

d,  2,983,  4,978,  9,399,  and  16,897  are  the  parts  that  make 
what  number  ? 

e.  83,469  added  to  what  number  will  make  103,497  ? 

2.  Find  the  least  number  that  will  exactly  contain  each  of 
these  numbers:  28,  35,  20,42. 

3.  What  is  the  number  whose  prime  factors  are  2,  3,  5,  7, 
11  and  17? 

4.  How  many  times  is  the  G.  C.  D.  of  48,  36,  and  72  con- 
tained in  their  L.  C.  M.  ? 

5.  Reduce  to  common  fractions  in  simplest  form:   a.    .025  ; 
h.  .375;    c.   .3125;    d.   .8375;    e.   .1275;  /.   .30;   g.   .625. 

6.  Reduce  -^^  to  a  decimal,  and  write  the  answer  in  words. 

7.  The  product  of  three  numbers  is  l^^.     Two  of  the  num- 
bers are  2^  and  1^^.     Find  the  other  number. 

8.  Which  is  larger,  and  how  much,  |  of  64  or  |^  of  45  ? 

7-^ 

9.  Change  -^  to  its  simplest  form. 

8 

10.  I  of  72  is  if  of  what  number  ? 

11.  Mrs.  Hill's  new  curtains  cost  $85.40.  This  was  f  as 
much  as  her  carpets  cost.  What  was  the  cost  of  both  curtains 
and  carpets  ? 


226 


REVIEW  AND  PRACTICE 


12.  In  a  certain  year  200,000  typewriting  machines  costing 
$12,500,000  were  made  in  the  United  States  by  10,000  men. 

a.    What  was  the  average  cost  of  the  machines  ? 

h.  At  the  average  rate,  how  many  men  were  needed  to 
conduct  a  factory  that  turned  out  20,000  machines  in  a 
year  ? 

<?.  If  the  labor  was  all  but  40  %  of  the  cost  of  the  machines, 
what  was  the  expense  for  labor  in  making  327  machines  ? 

d.  1440  of  these  machines  were  shipped  to  Mexico.  If 
they  were  sold  in  Mexico  at  an  average  price  of  $95  apiece, 
how  much  more  did  they  bring  than  the  cost  of  manufacture  ? 

e.  One  of  the  machines  in  the  picture  has  76  keys,  and 
the  other  two  have  each  42  keys.  If  each  of  the  three  factories 
where  these  machines  are  made  can  turn  out  50  complete  ma- 
chines in  a  day,  how  many  key  tops  are  needed  to  supply  the 
three  factories  for  one  week  ? 


13.  Find  the  areas  of  triangles  having  the  following  dimen- 
sions : 

a.  Base  28  rd.,  alt.  46  rd.  c.    Base  16  in.,  alt.  42  in. 

b.  Base  19  yd.,  alt.  23  yd.  d.    Base  64  ft.,  alt.  47 J  ft. 

14.  Find  the  altitude  of  a  triangle  whose  base  is  18  ft.  and 
whose  area  is  72  sq.  ft. 

15.  Find  the  base  of  a  triangle  whose  area  is  600  sq.  rd.  and 
whose  altitude  is  30  rd. 


REVIEW  AND  PRACTICE 


227 


16.  This  gable  is  covered  with  shingles 
that  cost  $5.50  per  M.  If  8  shingles  cover 
a  square  foot,  what  did  the  shingles  cost  ? 

17.  What  are  the  weekly  wages  of  a  girl 
who  finishes  4800  buttonholes  a  day,  if  she 
receives  1^  cents  per  dozen  garments,  and 
each  garment  contains  three  buttonholes  ? 

18.  John  Milton  was  born  Dec.  9,  1608, 
and  died  Nov.  8,  1675.     What  was  his  age  at  his  death  ? 

19.  The  battle  of  Bull  Run  was  fought  July  21, 1861.     How 
long  ago  was  that  ? 


Wheat  Blockade  in  the  Northwest 

20.  In  two  years  the  United  States  exported  154  million 
bu.  of  wheat. 

a.  In  how  many  days  would  1000  threshing  machines  thresh 
this  wheat  if  each  machine  threshes  1375  bu.  per  day  ? 

h.  How  many  grain  cars  would  carry  this  wheat  if  a  car 
can  carry  700  bushels  ? 

c.  If  a  bushel  of  wheat  weighs  60  lb.,  how  many  tons  would 
each  carload  weigh  ? 

d.  Allowing  33  ft.  for  the  length  of  each  car,  a  train  must 
be  how  long  to  carry  the  entire  quantity  of  wheat  ? 


228 


REVIEW  AND  PRACTICE 


Harvesting  Wheat 


21.  48  %  of  the  yield  of  wheat  in  a  western  township  in  one 
year  amounted  to  73,728  bu. 

a.    What  was  the  entire  yield  of  this  township  ? 

h.  If  the  average  yield  was  20  bu.  per  acre,  and  50%  of 
the  land  was  sown  to  wheat,  how  many  acres  of  land  were  there 
in  the  township  ? 

c.    If  the  township  was  a  rectangle  6  mi.  long,  how  wide  was  it  ? 

22.  How  many  days  elapsed  from  President  McKinley's  sec- 
ond inauguration,  March  4, 1901,  to  his  death,  Sept.  14,  1901  ? 

23.  The  distance  from  Covington,  Ky.,  to  New  York  is  996 
miles.  If  you  leave  Covington  at  1.30  p.m.  (New  York  time), 
May  1,  and  travel  toward  New  York  at  an  average  rate  of 
30  mi.  per  hour,  at  what  time  will  you  arrive  at  New  York  ? 

24.  How  many  j^ards  of  carpet  32  in.  wide  must  be  bought 
for  a  room  18'  x  16'  ? 

25.  Find  the  cost  of  the  paper  at  22  ^  a  roll  for  the  four  walls 
of  a  room  that  is  9  ft.  6  in.  high,  21  ft.  long,  and  15  ft.  wide, 
making  allowance  for  a  baseboard  9  in.  wide  all  around  the 
room,  and  for  two  doors  and  three  windows,  each  8  ft.  by 
7J  ft.     Estimate  a  roll  of  paper  to  cover  30  sq.  ft. 


REVIEW  AND   PRACTICE 


229 


26.  Henry  was  invited  to  his  uncle's  camp  in  the  Adirondack 
Mountains  for  a  two  weeks'  vacation.  He  had  saved  |20  from 
his  earnings  during  the  year. 

a.  With  a  part  of  this  money  he  purchased  the  following 
articles  : 

1  Paragon  bait  rod,  12.50. 
1  multiplying  reel,  11.40. 
1  enameled  silk  line,  $.90. 
3  doz.  snelled  hooks  at  $.35  per  dozen. 
1  canvas  hook  and  tackle  book,  |.50. 
1  willow  trout  basket,  f  1.10. 
How  much  did  these  articles  cost  ? 

I.  He  bought  a  railroad  mileage  book  of  500  miles  for  |10. 
From  this  he  paid  his  railroad  fare  for  98  miles  each  way. 
How  many  mile  slips  were  left  in  the  book  ? 

c.  What  did  Henry's  fare  cost  ? 

d.  He  paid  1.75  for  his  ride  in  a  buckboard  wagon  from  the 
station  where  he  left  the  cars  to  his  uncle's  camp,  a  distance  of 
6  miles.     What  was  the  rate  per  mile  ? 


230 


REVIEW   AND  PRACTICE 


e.  He  purchased  of  g.  St.  Regis  Indian  woman  for  60  cents 
a  sweet-grass  workbasket  for  his  mother,  and  for  40  cents  a 
handkerchief  box  for  his  sister.  His  expenses,  other  than  those 
mentioned,  amounted  to  ^t)  cents.  How  much  money  did 
Jlenry  have  when  he  returned  home  ? 

/.  What  was  the  remainder  of  his  mileage  book  worth,  at  the 
same  rate  per  mile  at  which  he  bought  it  ? 

g.    What  was  the  entire  cost  of  his  vacation  ? 

A.  Henry  and  his  uncle  went  trout  fishing  eight  times. 
Their  catches  for  the  different  trips  weighed  as  follows  ;  3  lb. 
5  oz.,  2  lb.  5  oz.,  7  lb.  4  oz.,  1  lb.  12  oz.,  2  lb.  2  oz.,  5  lb.  4  oz., 
10  oz.,  4  lb.  8  oz.     What  was  the  total  weight  ? 

i.  If  the  weight  of  the  trout  averaged  7  oz.  apiece,  how 
many  did  they  catch  ? 

27.  I  of  a  field  is  planted  with  corn  and  \  is  sown  with  wheat. 
The  corn  occupies  how  many  times  as  much  land  as  the  wheat? 

28.  What  is  the  value  of  6  acres  of  land  when  4  acres  of  the 
same  land  are  worth  $420? 

29.  If  a  ship  has  water  enough  to  last  25  men  8  months, 
how  long  will  the  water  last  8  men? 


INTEREST  231 


INTEREST 


274.  When  we  have  the  use  of  property  belonging  to  another, 
we  pay  the  owner  for  the  privilege  of  using  it.  For  instance, 
if  Mr.  A  lives  in  a  house  that  belongs  to  Mr.  B,  he  pays  Mr. 
B  a  certain  sum  for  the  use  of  the  house.  That  sum  is 
called  what  ?  The  amount  that  Mr.  A  pays  depends  upon 
the  value  of  the  house  and  the  length  of  time  for  which  the 
rent  is  paid. 

When  we  rent  a  horse  and  carriage  from  a  liveryman,  we  pay 
for  their  use.  The  amount  which  we  pay  depends  upon  the 
kind  of  horse  and  carriage,  the  length  of  time  we  use  them,  the 
distance  we  drive,  etc. 

When  you  rent  a  boat  at  the  boat  livery,  you  pay  for 
it  according  to  the  time  you  use  it.  Can  you  give  other 
illustrations  of  paying  for  the  use  of  property  belonging  to 
others  ? 

Sometimes  a  man  finds  it  necessary  to  borrow  money  from 
another.  Can  he  return  the  exact  pieces  of  money  which  he 
borrowed  ?  Should  he  return  just  as  much  as  he  borrowed  ? 
Should  he  return  more  than  he  borrowed  ?      Why  ? 

The  sum  paid  for  the  use  of  a  large  quantity  of  money  should 
compare  how  with  the  sum  paid  for  the  use  of  a  smaller  quan- 
tity of  money  ? 

The  sum  paid  for  the  use  of  some  money  for  a  long  time 
should  compare  how  with  the  sum  paid  for  the  use  of  the  same 
money  for  a  short  time  ? 

The  price  paid  for  the  use  of  the  same  quantity  of  money 
for  the  same  time  varies  in  different  places. 

275.  Money  paid  for  the  use  of  money  is  interest. 

276.  Money  for  the  use  of  which  interest  is  paid  is  the  prin- 
cipal. 


232  INTEREST 

277.  The  sum  of  the  principal  and  interest  is  the  amount. 

278.  The  sum  to  he  paid  for  the  use  of  money  is  always  de- 
termined hy  taking  a  certain  per  cent  of  the  principal. 

279.  The  number  of  hundredths  of  the  principal  taken  as  the 
interest  for  one  year  is  the  rate  of  interest.  For  instance,  if  *„ 
sum  of  money  is  borrowed  and  6%  of  that  sum  is  the  interest 
for  one  year,  the  rate  of  interest  is  6  % . 

280.  ^-Phe  rate  of  interest  which  is  fixed  hy  law  is  called 
the  legal  rate.  In  a  majority  of  the  states  the  legal  rate  is 
6%.  In  some  states  it  is  greater  than  6  %,  and  in  some  states 
less. 

A  lower  rate  than  the  legal  rate  is  always  allowed  by  law  if 
the  debtor  and  creditor  so  agree.  In  a  few  states  a  higher  rate 
than  the  legal  rate  is  allowed  if  the  debtor  g,nd  creditor  so 
agree  ;  but  in  most  states  a  higher  rate  than  the  legal  rate  is 
forbidden  by  law.     What  is  the  legal  rate  where  you  live  ? 

281.  Interest  at  a  higher  rate  than  that  permitted  hy  law  is 
usury. 

282.  Oral 

1.  Mr.  Smith  borrowed  from  Mr.  Arnold  $100  for  1  yr. 
*At  the  end  of  the  year  Mr.  Smith  repaid  the  money  which  he 
had  borrowed  and  also  paid  Mr.  Arnold  6  %  interest.  How 
much  was  the  interest  ?  What  was  the  principal  ?  How  much 
did  Mr.  Arnold  receive  in  all  ?  What  is  this  sum  called  ? 
Who  was  the  debtor  ?     Who  was  the  creditor  ? 

2.  What  is  the  interest  on  1500  at  6%  fori  yr.?  On  $  800  ? 
On  I  900?      On  $300?     On  $1000?     On  $250? 

3.  What  is  the  interest  on  $500  for  1  yr.  at  5%  7  At  10%  ? 
At7%?    At4%?    At3%?    At8%? 


INTEREST  233 

4.  What  is  the  interest  on  ilOOO  at  5%  for  1  yr.  ?     For 
2  yr.  ?     For  3  yr.  ?     For  8  yr.  ?     For  10  yr.  ? 

5.  What  is  the  interest  on  1 100  for  2  yr.  at  6fo  ?     At  4  %  ? 


At3%?     At9%?     At8%? 

6.  Six  months  are  what  part  of  a  year  ?  3  mo.  ?  4  mo.  ? 
8  mo.  ?     9  mo.  ?     10  mo.  ?     1  mo.  ? 

7.  What  is  the  interest  on  $  600  for  1  yr.  at  6  %  ?  For  6 
months?  For  3  mo.  ?  For  4  mo.  ?  For  8  mo.?  For  9  mo.  ? 
For  1  mo.  ?     For  11  mo.  ? 

283.     Written 

1.    What  is  the  interest  on  $2000  for  3  yr.  at  7  %  ? 

x-tL-x|-=|420    Ans, 


1      m    1 

How  do  we  find  the  interest  for  1  yr.  ?    For  3  yr.  ? 

2     What  must  be  paid  for  the  use  of  $700  for  4  mo.  at  6% 
per  year  ? 

2 

3.  Find  the  interest  on  $350  for  3  yr.  6  mo.  at  4%.     3  yr. 
6  mo.  =  how  many  years  ? 

7          ? 
iMx-i-xI-$ Ans 

4.  JPmi  ^^e  interest  on  $800  a^  6  %  : 

a.    For  7  yr.  '  ^.    For  21  yr.  <?.    For  9  mo. 


234  INTEREST 

d.  For  2  yr.  6  mo.      /.    For  5  mo.  h.    For  2  yr.  10  mo. 

e.  For  1  yr.  8  mo.      g.    For  1  yr.  5  mo.       ^.    For  3  yr.  7  mo. 

Note.  — In  final  results,  a  part  of  a  cent  smaller  than  |  cent  is  dropped ;  a 
fraction  equal  to,  or  greater  than,  \  cent  is  called  one  cent. 

5.  Find  the  interest  on  il600  at  b%  : 

a.  For  5  yr.  c.  For  2|  yr.  e.    For  1  yr.  3  mo. 

h.  For  4 J  yr.  d.  For  3  yr.  8  mo.        /.    For  2  yr.  3  mo. 

6.  Find  the  interest  on  |400  for  3  yr.  at  31  %. 

7.  TF^a^  ^s  fAe  interest: 

a.  On  $200  for  4  yr.  at  5  %  ? 

5.  On  $1400  for  1  yr.  at  ^  %  ? 

c.  On  11800  for  6  mo.  at  6f  %  ? 

c?.  On  11500  for  3  yr.  4  mo.  at  4  %  ? 

e.  On  11200  for  2  yr.  8  mo.  at  4|  %  ? 

284.  It  is  the  custom  of  business  men  in  computing  interest 
to  consider  one  month  as  30  days,  and  one  year  as  360  days. 

Allowing  360  days  for  a  year,  100  days  are  what  part  of  a 
y^ar  ?  200  da.?  150  da.?  10  da.?  75  da.?  90  da.? 

1.    What  is  the  interest  on  $720  at  6  %  for  100  da.? 

2 


e. 

For  90  da. 

/. 

For  45  da. 

{. 

For  33  da. 

J'. 

For  93  da. 

k. 

For  m  da. 

I. 

For  345  da 

INTEREST  285 

2.  Find  the  interest  on  $720  at  Q%: 
a.  For  200  da.  c.  For  10  da. 
h.    For  150  da.                   d.    For  75  da. 

3.  Compute  the  interest  on  $1800  at  i%: 

a.  For  200  da.  e.  For  45  da. 

b.  For  150  da.  /.  For  75  da. 

c.  For  90  da.  g.  For  110  da. 

d.  For  30  da.  h.  For  240  da. 

4.  Compute   the   interest  on  $75  at  4%  for  2  mo.  20  da. 
2  mo.  20  da.  =  how  many  days  ?     How  many  years  ? 

^  2 

3 

5.  Find  the  interest  on: 

a.  $280  for  3  mo.  10  da.  at  3  %. 

h.  $375  for  7  mo.  9  da.  at  8%. 

c.  $500  for  11  mo.  6  da.  at  9  %. 

d.  $450  for  5|mo.  at  8%. 

e.  $300  for  4  mo.  12  da.  at  7|%. 

6.  What  is  the  interest  on  $  450  at  6  %  for  1  yr.  1  mo.  10 

da.? 

1  yr.  1  mo.  10  da.  =  360  da.  +  30  da.  +  10  da.,  or  400  da. 

5  ^ 


236  INTEREST 

7.  Compute  the  interest  on: 

a,  1300  for  1  yr.  5  mo.  12  da.  at  5%. 

h.  1900  for  1  yr.  7  mo.  11  da.  at  4  %. 

c.  $360  for  1  yr.  2  mo.  7  da.  at  7  %. 

d,  $840  for  2  yr.  15  da.  at  6  %. 

8.  Compute  the  interest  on  $660  for  2  yr.  8  mo.  at  7|^5^, 
71       15 


'^2^' "100  "200 


3.30        5        8 

xlixll  =  $132   AnB, 


m  n 
m   i 

Note.  —  We  cancel  100,  then  divide  330  by  100  by  pointing  ofE  two  deci- 
mal places. 

9.    Find  the  interest  on  175.25  at  8%  for  20  da. 
3.01  ^ 

9 

10.    Compute  the  interest  on : 

a.  $485.50  for  1  yr.  3  mo.  at  4  %. 

h.  $125.50  for  1  yr.  4  mo.  at  41  %. 

c,  $  240  for  8  mo.  15  da.  at  5^  %. 

d,  $540  for  1  yr.  4  mo.  10  da.  at  5  %. 

From  the  foregoing  examples  we  observe  that  the  interest  on 
any  sum  of  money,  at  any  rate,  for  any  time,  is  always  the  prod- 
uct of  three  factors.     What  are  they  ? 

In  what  denomination  is  the  time  expressed  before  multiply- 
ing to  find  the  interest? 


INTEREST  237 


285.    Written 

In  examples  1-24  compute  the  interest. 


Principal 

Rate 

Time 

1. 

14320 

Hfo 

2  yr.  6  mo. 

2. 

1720 

5% 

2  yr.  8  mo. 

11  da. 

3. 

$1081.08 

7% 

6  mo. 

20  da. 

4. 

$5000 

Hfo 

2  yr.  8  mo. 

5. 

1901.80 

8% 

3yr. 

24  da. 

6. 

$1236.48 

H% 

2  yr.  2  mo. 

2  da. 

7. 

1620.40 

5|% 

2  yr.  3  mo. 

10  da. 

8. 

I12T5.30 

9% 

5  yr.  6  mo. 

15  da.^ 

9. 

11500 

61% 

2  yr.  9  mo. 

9  da. 

10. 

$270.27 

6% 

2yr. 

27  da. 

11. 

$396 

10% 

1  yr.  1  mo. 

9  da. 

12. 

$444 

4% 

5  yr.  6  mo. 

13. 

$84.50 

7% 

2  yr.  5  mo. 

12  da. 

14. 

$16.75 

7% 

7  mo. 

17  da. 

15. 

$336 

5% 

15  da. 

16. 

$300.50 

3% 

1  yr.  2  mo. 

15  da. 

17. 

$42.20 

4i% 

lyr. 

16  da. 

18. 

$51.17 

4% 

9  mo. 

29  da. 

19. 

$35.50 

7% 

3  yr.  5  mo. 

20  da. 

20. 

$691.04 

5% 

1  mo. 

3  da. 

21. 

$640.50 

10% 

10  mo. 

26  da. 

22. 

$105.10 

12% 

48  da. 

23. 

$92.96 

7% 

4  mo. 

3  da. 

24. 

$31.40 

7% 

273  da. 

238  INTEREST 

25.  Mr.  Ward  borrowed  1 10,000  of  Mr.  Beach  at  5  %  and 
lent  it  to  Mr.  Waite  at  6  %.  Mr.  Waite  kept  the  money  2  yr. 
6  mo.  18  da.  He  then  paid  Mr.  Ward,  and  Mr.  Ward  paid 
Mr.  Beach.     What  was  Mr.  Ward's  gain  ? 

26.  What  must  be  paid  for  the  use  of  $  160  for  11  yr.  11 
mo.  11  da.  at  7i 


70 


27.  Required,  the  interest  on  $  2000  for  3  yr.  7  mo.  12  da. 
at  3  %. 

28.  How  much  will  f  358.50  earn  in  1  yr.  8  mo.  6  da.  when 
put  on  interest  at  7  %  ? 

29.  How  much  interest  will  $475.50  earn  in  5  yr.  9  mo.  24 
da.  at  4  %  ? 

30.  How  much  interest  will  $840.50  yield  in  10  mo.  15  da. 
at  41  %  ? 

31.  At  5  %  interest,  what  will  $  75  gain  in  11  mo.  10  da.? 

286.     Oral 

1.  The  sum  of  the  principal  and  interest  is  called  what  ? 

2.  Mr.  Williams  borrowed  $  100  of  Mrs.  Johnson,  paying  6  % 
interest.     What  was  the  principal  ?  The  interest  ?  The  amount  ? 

3.  Name  the  principal,  interest,  and  amount  when  $  100  is 
borrowed  for  3  yr.  at  6%  interest. 

4.  Find  the  amount  of  $  50  for  IJ  yr.  at  6%. 

5.  Find  the  amount  of  1 1  for  7  yr.  at  5%. 

6.  What  is  the  amount  of  $  100  for  6  mo.  at  5%  ? 

7.  If  I  borrow  $  200  at  4%  interest,  how  much  do  I  pay  in 
three  years  ? 

8.  Frank's  uncle  gave  him  a  New  Year's  present  of  $  100, 
which  he  put  in  a  bank  that  paid  4%  interest.  How  much 
money  did  Frank  have  in  the  bank  at  the  end  of  three  months  ? 


INTEREST  239 

9.    What  is  the  amount  of  $  300  for  four  years  at  5%  ? 

10.  A  man  bought  a  span  of  horses  for  $  150  apiece  and  a 
wagon  for  $  50,  agreeing  to  pay  for  them  in  one  year  with  inter- 
est at  10%.     How  much  did  he  have  to  pay  ? 

11.  Mr.  B  owes  $  1000  on  his  house  and  pays  the  interest 
every  six  months  at  the  rate  of  5%  per  year.  How  much  inter- 
est does  he  pay  each  time  ? 

12.  I  owe  a  debt  of  $  40,  due  in  one  year  and  six  months.  If 
I  pay  interest  at  the  rate  of  10%  per  year,  how  much  will  the 
debt  amount  to  when  it  is  due  ? 

13.  What  is  the  interest  on  $  200  for  five  years  when  the  rate 
of  interest  is  3J%  ? 

14.  What  is  the  amount  of  1 400  for  two  years  at  3^%  ? 

15.  The  amount  of  $400  for  a  certain  time  at  7  %  interest 
is  1428.     Can  you  find  the  time  ? 

16.  The  amount  of  1 100  for  2  years  is  f  110.  Can  you  find 
the  rate  of  interest  ? 

17.  The  amount  of  a  sum  of  money  for  one  year  at  7%  inter- 
est is  1214.     Can  you  find  the  principal  ? 

18.  The  interest  of  1 50  for  2  years  is  1 6.  Can  you  find  the 
rate  of  interest  ? 

19.  The  amount  of  $80  for  1^  years  at  5%  per  year  is  how 
much? 

20.  When  the  principal,  rate,  and  time  are  given,  how  may 
the  amount  be  found  ? 

21.  The  principal  added  to  the  interest  gives  what  ? 

22.  The  difference  between  the  principal  and  amount  is  what  ? 
The  difference  between  the  interest  and  amount  ? 


240  INTEREST 

287.     Written 

In  examples  1~1S  find  the  amount: 


Principal 

Eatb 

Time 

1. 

1450 

6% 

1  JT, 

2  mo. 

20  da. 

2. 

$150.50 

8% 

20  da. 

3. 

1330 

71% 

2yr. 

8  mo. 

4. 

17500 

4% 

2  mo. 

20  da 

5. 

11250 

4|% 

2yr. 

8  mo. 

6. 

1901.80 

8% 

lyr. 

6  mo. 

12  da. 

7. 

1620.40 

61% 

lyr. 

1  mo. 

20  da. 

8. 

1444.60 

3% 

lyr. 

6  mo. 

9. 

$42.25 

7% 

lyr. 

5  mo. 

12  da. 

10. 

I960 

'  6% 

11  mo. 

20  da. 

U. 

1173 

6% 

8  mo. 

16  da. 

12. 

$1500 

8% 

2yr. 

5  mo. 

13  da. 

13. 

$90 

6f% 

lyr. 

27  da. 

14. 

What  will  $1000  amount  to  in 

1  yr.  7 

mo.  if 

put  at 

intere^ 

3t  at  4%? 

15.  What  will  $8450  amount  to  in  90  da.  at  10  %  ? 

16.  A  man  borrowed  $416  at  5%.  If  nothing  was  paid  on 
the  debt  for  1  yr.  16  da.,  how  much  did  the  man  then  owe  ? 

17.  If  you  should  borrow  $150.25  at  the  legal  rate  of  inter- 
est where  you  live,  how  much  would  you  owe  11  mo.  15  dao 
after  borrowing  the  money? 

18.  If  a  man  borrows  $146.75  to-day  at  the  legal  rate  of 
interest  where  you  live,  how  much  will  he  owe  9  mo.  15  da. 
from  to-day? 

19.  P'ind  the  amount  of  $750.25  for  1  yr.  27  da.  at  9%. 


INTEREST  .  241 

PROBLEMS  IN  WHICH   THE  TIME  MUST  BE  COMPUTED 
BEFORE  INTEREST  CAN  BE  FOUND 

288.     Written 

1.  What  is   the  interest  on  |144  from   June   12,   1901,  to 
Jan.  2,  1903,  at4%? 

1903  yr.  1  mo.  2  da.  ^^ 

1901     6    12  ^"^m^m^^^'^^  ^'''* 

1        6       20  Difference  in  Thne  ^  ^ 

2.  Find  the  interest  on : 

a.  1500       at  6   %  from  May     7,  1902,  to  Sept.    7,  1904 

h.  172  at  41  %  from  Apr.     1,  1904,  to  Apr.   16,  1907. 

c.  $60  at  51%  from  Sept.  30,  1899,  to  June  15,  1901. 

d.  1240.60  at  10%  from  Oct.    25,  1904,  to  Dec.   10,  1907. 

e.  $360  at  9  %  from  Nov.  30,  1903,  to  May  25,  1905. 
/.  $1000  at  3|%  from  Jan.  21,  1904,  to  June  11,  1906. 
g.  $48.48  at  8  %  from  Feb.  25,  1906,  to  Jan.  5,  1908. 
h.  $99  at  6  %  from  Sept.  21,  1904,  to  Jan.  1,  1906. 
i.  $36.36  at  10 %  from  Feb.  2,  1903,  to  Oct.  22,1905. 
y.  $900  at  31  %  from  Dec.  30,  1904,  to  Jan.  15,1905. 
h.  $45.90  at  6    %  from  Jan.      9,  1900,  to  Aug.     5,  1903. 

I.  $576  at  4  %  from  July  4,  1902,  to  Feb.  3,  1905. 
m.  $960.84  at  3  %  from  Apr.  3,  1904,  to  Sept.  6,  1907. 
n.    $162.72  at  5   %  from  May   12,  1900,  to  May     4,  1905. 


242  INTEREST. 


INTEREST  FOR  SHORT   PERIODS 

289.     When  money  is  on  interest  for  less  than  a  year,  it  is 
customary  to  compute  the  time  in  days. 

What  is  the  interest  on  $1575.25  from  Jan.  9, 1904,  to  March 

15,  1904,  at  3%  ? 

r  22  da.  left  in  Jan. 
The  money  is  on  interest  for  \  29  da.  in  Feb. 

[  15  da.  in  March 
66  da.     Term  of  Interest 

787.625  11 

Wl^'M    xjgx  ^  =  18.663875,  or  $8.66    Am, 


10 

Observe  that  by  keeping  (not  cancelling)  the  factors  100  and  10  below  the 
line  we  may  multiply  together  the  factors  above  the  line  and  then  divide 
by  100  and  10  by  pointing  off  three  more  decimal  places  in  the  product. 

290.     Written 

1.  Compute  the  interest  on  1721.44  at  4%  from  April  3  to 
July  6. 

2.  Find  the  interest  on  $9000  at  5%  from  March  4,  1898,  to 
April  3,  1898. 

3.  A  man  borrowed  $576.72,  May  12,  1896.  How  much 
did  he  owe  Aug.  10,  1896,  computing  the  interest  at  5  %  ? 

4.  A  man  borrowed  $3500,  Jan.  5,  1902,  and  repaid  the 
money  with  interest  at  6%  on  April  1,  1902.  How  much  did 
he  pay  ?  " 

5.  Find  the  amount  of  $250  borrowed  June  1,  1907,  and 
paid  Aug.  21,  1907,  with  interest  at  6%. 


INTEREST  248 

6.  Find  the  interest  on : 

a.  1600  from  Aug.  1  to  Aug.  21,  1907,  at  51%. 

h.  1120  from  May  6  to  May  31,  1905,  at  5%. 

e.  1219  at  7  %  from  June  12  to  July  30,  1904. 

d.  1638  at  6%  from  Dec.  15,  1906,  to  Feb.  8,  1907. 

e.  $1000  at  7|  %  from  Nov.  18,  1907,  to  Feb.  17,  1908. 
/.  1248.50  at  9%  from  Aug.  15  to  Aug.  31,  1901. 

^.  1631.78  at  10  %  from  Jan.  14  to  Sept.  8,  1896. 

h.  148.70  at  10%  from  May  3  to  Oct.  7,  1899. 

i,  $246.42  at  8%  from  Sept.  30,  1907,  to  Jan.  1,  1908. 

y.  1401.28  at  4|  %  from  July  8,  1906,  to  Jan.  1,  1907. 

k.  $283.49  at  6%  from  Dec.  31,  1902,  to  March  30,  1903. 

I  $800  at  5%  from  Dec.  1,  1903,  to  April  1,  1904. 

m.  $12,000  at  61%  from  Jan.  30  to  June  16,  1896. 

n.  $200,000  at  3f  %  from  Aug.  5  to  Aug.  23,  1905. 

0.  $150,000  at  41  %  from  March  10  fo  July  18,  1907. 

p.  $76.47  at  9%  from  April  1  to  April  29,  1901. 

q.  $84.13  at  7  %  from  Feb.  20  to  Aug.  1,  1907. 

r.  $43,475  at  41%  from  May  7,  1899,  to  Jan.  14,  1900. 

8.  $4376.40  at  6%  from  May  6,  1900,  to  March  17,  1901. 

7.  Find  the  amount  of: 

a.  $400  at  7  %  from  Aug.  31  to  Dec.  1,  1904. 

h.  $308.12  at  6  %  from  Feb.  1  to  March  13,  1906. 

c.  $242.14  at  7  %  from  Jan.  31  to  April  3,  1904. 

d.  $800,000  from  June  16,  1900,  to  Jan.  1,  1901,  at  31%. 

e.  $140,000  from  April  14  to  May  19,  1904,  at  4|  %. 

/.  $131.13  at  8%  from  Oct.  31,  1905,  to  Feb.  27,  1906. 

g.  $434.25  at  5i%  from  Feb.  21  to  July  3,  1907. 


244  REVIEW  AND  PRACTICE 

291.    Oral  REVIEW  AND  PRACTICE 

1.  Expressing  numbers  by  means  of  figures  is  called  what  ? 

2.  Name  the  first  six  places  in  integers. 

3.  Define  an  integer. 

4.  ReadMCMIX. 

5.  Numerate  20756.3010. 

6.  The  numbers  added  are  called  what? 

7.  Find  the  value  of  8  +  3  x  2  -  21  -5-  7. 

8.  Giive  the  sums  rapidly  : 

18  +  8;  27  +  12;  26  +  19;  49  +  48;  53  +  47. 

9.  How  may  we  test  our  work  in  subtraction? 

10.  How  may  we  test  our  work  in  multiplication  ? 

11.  How  may  we  test  our  work  in  division? 

12.  Define  division. 

13.  Which  terms  in  division  are  factors? 

14.  Which  terms  in  multiplication  are  factors? 

15.  Grive  results  rapidly : 

43-12;  56-17;  93-56;  85-59;  132-94. 

16.  Multiply  hy  100  : 

48;  4264;  408;  37.9;  84.729;  .0079.    ^ 

17.  Multiply  48  by  25. 

18.  Divide  hy  1000  : 

2645.3;  793;  4835;  3.9;  9638.2;  7;  82. 

19.  When  20  pickles  cost  13  cents,  how  much  should  be  paid 
for  80  pickles? 

20.  ^At  the  rate  of  7  for  3  cents,  how  many  screw  hooks  can 
be  bought  for  15  cents  ? 


REVIEW  AND  PRACTICE  245 

21.  How  much  a  dozen  is  paid  for  bananas  when  36  bananas 
cost  60  cents  ? 

22.  Name  the  odd  numbers  between  40  and  50. 

23.  Name  the  prime  numbers  from  1  to  47. 

24.  Name  the  prime  factors  of  60. 

25.  What  number  will  divide  every  even  number  ? 

26.  How  may  we  tell  whether  a  number  is  prime  or  not  ? 

27.  What  is  cancellation  ? 

28.  The  smallest  number  that  exactly  contains  each  of  two 
or  more  numbers  is  called  what  ? 

29.  What  is  the  greatest  number  that  will  exactly  divide  45 
and  60  ? 

30.  Find  the  L.  C.  M.  and  G.  C.  D.  of  36  and  8. 

31.  A  fraction  is  always  an  expression  of  what  operation  ? 

32.  Which  term  of. a  fraction  is  the  dividend? 

33.  Which  term  of  a  fraction  is  the  divisor  ? 

34.  Which  is  greater,  J  or  J?  |  or  |  ?  f  or  f  ?  ^3_  or  ^ ? 

35.  The  "  Lincoln  Stars  "  won  5  games  of  baseball,  with  the 
following  scores  :  7,  8,  2,  9,  4.     What  was  their  average  score  ? 

36.  Name  five  aliquot  parts  of  one  dollar. 

37.  How  many  packages  of  cereal   at  1.12^  each   can   be 
bought  for  13? 

38.  What  per  cent  of  anything  is  J  of  it  ?  ^  of  it  ?  J  ?  |  ?  -J  ? 
f?|?|?f? 

39.  If  the  interest  on  $25  for  a  certain  time  is  $3,  what  is 
the  interest  on  $200  for  the  same  time,  at  the  same  rate  ? 

40.  62 J  %  of  f  16  is  how  much  money  ? 

41.  Draw  a  line  one  yard  long  without  a  measure.     Measure 
and  correct  it. 


246  REVIEW  AND  PRACTICE 

42.  Draw  a  square  foot.     Measure  and  correct  it. 

43.  Draw  a  square  8  inches  on  a  side.  Measure  and  cor- 
rect it. 

44.  How  many  days  are  there  in  the  summer  months  ? 

45.  Each  spoke  in  a  certain  wheel  makes  an  angle  of  60° 
with  the  spoke  next  to  it.  How  many  spokes  are  there  in  the 
wheel  ? 

46.  When  a  stationer  buys  tablets  at  the  rate  of  $6  a  gross, 
how  much  does  he  pay  for  each  dozen  ?  When  he  buys  them 
at  ^  9  a  gross,  what  does  he  pay  for  each  dozen  ? 

47.  I  bought  a  ream  of  note  paper  and  used  12  quires  of  it. 
How  many  sheets  were  left  ? 

48.  One  circumference  is  equal  to  how  many  arcs  of  ten 
degrees  each  ? 

49.  From  the  Fourth  of  July  to  Christmas  is  how  many 
days  ? 

50.  What  is  the  cost  of  2500  shingles  at  $6  per  M.? 

51.  What  is  the  area  of  a  triangle  whose  base  is  20  inches 
and  altitude  is  1  foot  ? 

52.  What  is  the  altitude  of  a  triangle  whose  base  is  12  feet 
and  whose  area  is  48  square  feet  ? 

53.  How  many  feet  of  lumber  are  there  in  a  board  which  is 
14  ft.  long,  6  in.  wide,  and  |  in.  thick  ? 

54.  What  is  the  amount  of  1300  for  1^  yr.  at  3  %  ? 

55.  The  interest  on  a  sum  of  money  for  ten  months  is  $  35. 
What  is  the  interest  on  the  same  sum,  at  the  same  rate,  for  two 
months  ? 

56.  1^  of  a  flag  pole  126  ft.  high  was  broken  off  by  the  wind. 
How  many  feet  high  was  the  piece  left  standing  ? 


REVIEW   AND  PRACTICE 


247 


292.    Written 

1.  Express  in  Roman  numerals  the  number  of  the  year  in 
which  you  were  born. 

2.  Write  in  words  500200.00202. 


I.  Add,  and  test 

your  work : 

a 

b 

c 

d 

358 

26 

98034 

12843 

47 

544 

576 

798 

968 

37 

934 

6347 

7684 

829 

86 

999 

9235 

5444 

715 

3857 

1386 

308 

8397 

92124 

428 

9176 

869 

28315 

•79 

283 

476 

76543 

9830 

706 

59 

89412 

49 

2493 

987 

6347 

15730 

819 

43 

48009 

2132 

6478 

3754 

90384 

4.  The  sum  of  two  numbers  is  80,305.  One  of  the  addends 
is  79,496.     Find  the  other. 

5.  One  rectangular  field  is  35  rods  by  54  rods ;  another  is 
24  rods  by  51  rods.  How  many  more  feet  of  fence  are  required 
to  inclose  one  of  them  than  to  inclose  the  other  ? 

6.  Mr.  Walch  sold  two  horses  for  $275  each,  and  a  carriage 
for  i295.  He  then  bought  an  automobile  costing  four  and 
three  fourths  times  as  much  as  the  horses  and  carriage  brought. 
He  received  how  much  less  than  he  paid  out?  Indicate  the 
work,  and  then  find  the  answer. 

7.  Make  and  solve  a  problem  that  requires  the  following 
operations:   16  x  8.45+ 84  x  $.70-132.18. 


248  REVIEW  AND  PRACTICE 

8.  5.375-^(1.55 -.3) +  142.34x7  =  ? 

9.  Find  the  prime  factors  of  589. 

10.  Find  the  smallest  number  that  will  exactly  contain  each 
of  the  numbers  105,  56,  84,  220. 

11.  Find  the  largest  number  that  will  exactly  divide  260, 
490,  1078,  364. 

12.  One  factor  of  f f  is  ||.     Find  the  other. 

13.  Divide  3^  of  f  by  If  of  If . 

14.  Divide  I  of  f  by  l|  of  if. 

15.  Find  the  value  of  f  X  4|  divided  by  ^  of  8f 

16.  Change  i^rYto  to  a  fraction  whose  terms  are  prime  to 
each  other. 

17.  How  much  greater  is  |  of  41  than  51^j  -  42i|? 

18.  What  decimal  is  exactly  equal  to  ||^? 

19.  Mr.  Tripp  kept  a  record  of  the  temperature  indicated  by 
his  thermometer  at  noon  every  day  for  a  week  as  follows :  76°, 
80°,  78°,  83°,  87°,  90°,  89°.  What  was  the  average  noon 
temperature  for  the  week? 

20.  Make  out  a  bill  of  four  items,  using  prices  found  in  the 
newspaper.  Receipt  the  bill  as  if  you  were  the  creditor's 
clerk. 

21.  Multiply  43.761  by  5.8|. 

22.  Divide  74.96f  by  .6^. 

23.  Multiply  3867  by  25,  in  the  shortest  way. 

24.  Divide  3976  by  25,  in  the  shortest  way. 

25.  A  contractor  agreed  to  excavate  a  cellar  180  ft.  by  45  ft. 
and  11  ft.  deep.  What  fraction  of  the  work  was  left  undone 
when  he  had  removed  175  cu.  yd.  of  earth  ? 

26.  What  common  fraction  is  the  same  as  11|  %  ? 


REVIEW  AND  PRACTICE 


249 


27.  15  %  of  the  weight  of  a  certain  piece  of  cloth,  was  wool. 
If  the  piece  contained  25J  lb.  of  cotton,  how  many  pounds  did 
the  piece  weigh  ? 

28.  In  excavating  the  side  of  a  hill  .for  a  railroad,  it  was  nec- 
essary to  remove  8500  cubic  yards  of  clay  and  rock.  If  19|  % 
of  the  material  removed  was  rock,  how  many  cubic  feet  of  clay 
were  removed  ? 

29.  a.  A  brick  wall  41  ft.  6  in.  long,  1  ft.  6  in.  wide,  and 
8  ft.  high  contains  how  many  bricks,  if  22  bricks  will  make 
1  cu.  ft.  of  the  wall  ? 

h.   They  cost  how  much  at  19.50  per  M.  ? 

30.  16,200  bricks  were  required  in  building  a  wall  2  ft.  thick 
and  9  ft.  high.  If  it  required  24  bricks  to  make  a  cubic  foot 
of  wall,  how  long  was  the  wall  ? 

31.  a.  A  20-acre  vineyard  contains  540  vines  to  the  acre. 
All  the  vines  in  the  vineyard  are  set  in  90  equal  rows.  How 
many  vines  are  there  in  each  row  ? 


Vineyard 


250 


REVIEW  AND  PRACTICE 


h.  The  average  yield  is  1000  baskets  of  grapes  per  acre.  An 
empty  basket  weighs  IJ  lb,  A  filled  basket  weighs  8  lb.  How 
many  pounds  of  grapes  are  raised  on  an  acre  ? 

c.  How  many  tons  are  raised  in  the  whole  vineyard  ? 

d.  What  are  they  worth  at  824  per  ton  ? 

e.  If  it  costs  1  cent  per  basket  to  pick  and  pack  the 
grapes  and  y*^  of  a  cent  per  basket  for  cartage,  what  must 
be  paid  for  picking,  packing,  and  carting  an  acre's  yield  of 
grapes  ? 

32.  a.  Three  cents  a  tray  are  paid  for  picking  wine  grapes. 
What  are  the  weekly  wages  of  a  picker  who  picks  50  trays  of 
grapes  a  day  ? 

5.  If  64  filled  trays  weigh  a  ton,  and  the  yield  of  an  acre  is 
3J  tons,  including  the  weight  of  the  trays,  what  is  the  cost  of 
picking  three  acres  of  wine  grapes  ? 

c.  The  weight  of  an  empty  tray  is  5  lb.  What  is  the  weight 
of  the  grapes  that  fill  one  tray  ? 

d.  How  many  pounds  of  grapes 
will  fill  64  trays  ? 

e.  At  5  cents  apiece,  what  is 
the  cost  of  the  trays  for  an  acre 
of  grapes?    (See  5.) 

/.  At  one  cent  a  pound,  what 
is  the  value  of  the  grapes  from  an 
acre  of  ground  ? 

33.  a.  A  grape  grower  in  Cali- 
fornia has  40  acres  of  wine  grapes. 
It  costs  $12  an  acre  to  train  and 
cultivate  his  vines  and  11.35  per 
ton  for  picking.     If  the  yield  is 

6  tons  to  the  acre,  and  he  sells  the  entire  crop  for  $15  a  ton, 
what  is  bis  net  profit  per  acre  ? 


REVIEW  AND  PRACTICE  251 

h.  Each  ton  of  these  grapes  will  make  150  gallons  of  wine. 
Allowing  32  gallons  for  a  barrel,  how  many  barrels  of  wine  can 
be  made  from  the  entire  40-acre  vineyard  ? 

c.  How  many  pounds  of  grapes  are  used  in  making  one  gal- 
lon of  wine  ? 

34.  The  grapevines  are  supported  by  wires  fastened  to  posts. 
If  it  requires  609  lb.  of  wire  per  acre,  costing  842  a  ton,  what 
is  the  cost  of  the  wire  for  a  15-acre  vineyard  ? 

35.  What  will  it  cost  to  carpet  a  room  20  ft.  by  23  ft.  with 
carpet  27  in.  wide,  costing  11.75  a  yard,  with  8^  per  yard  added 
for  making  and  laying,  running  the  strips  the  longer  way  of  the 
room,  and  making  no  allowance  for  waste  in  matchipg  the  figure  ? 

36.  a.  What  will  it  cost  to  lath  and  plaster  the  walls  of  a 
room  80  ft.  by  30  ft.  and  14  ft.  high  at  40  cents  a  square  yard, 
making  full  allowance  for  20  windows,  each  3|^  ft.  by  7  ft.,  and 
4  doors,  each  3  ft.  3  in.  by  8  ft.? 

h.  The  floor  of  this  room  is  supported  by  120  joists,  each 
16  ft.  long,  12  in.  wide,  and  3  in.  thick.  What  did  they  cost 
at  $28  per  M.  board  feet.? 

37.  Find  the  sum  of  35°  46'  52''  and  72°  13'  38". 

38.  A  game  of  baseball  began  at  25  minutes  38  seconds  past 
2  P.M.,  and  closed  at  7  minutes  15  seconds  past  4  p.m.  How 
long  did  the  game  last  ? 

39.  The  widths  of  six  city  lots,  lying  side  by  side,  are  as  follows  : 
bb  ft.,  40  ft.  6  in.,  72  ft.  9  in.,  38  ft.  10  in.,  80  ft.,  and  ^  ft. 
8  in.     Find  in  feet  and  inches  the  entire  width  of  all  the  lots. 

40.  What  is  the  amount  of  1 350  at  interest  from  March  15, 
to  July  11,  1907,  the  rate  of  interest  being  5J  %  ? 

41.  Find  in  the  shortest  way  the  interest  on  a  sum  of  money 
for  30  days,  when  the  interest  on  the  same  sum,  at  the  same 
rate,  for  120  days,  is  137.16. 


TABLES 

LIQUID  MEASURE 

4  gills  (gi.)        =  I  pint  (pt.). 
2  pints  =  1  quart  (qt.). 

4  quarts  =  1  gallon  (gal.). 

DRY  MEASURE 

2  pints  (pt.)  =  1  quart  (qt.) . 
8  quarts  =  1  peck  (pk.). 
4  pecks  =  1  bushel  (bu.). 

AVOIRDUPOIS  WEIGHT 

16  ounces  (oz.)   =  1  pound  (lb.). 
2000  pounds  =  1  ton  (T.). 

2240  pounds  =  1  long  ton. 

100  pounds  =  1  hundredweight  (cwt.), 

LINEAR   MEASURE 

12  inches  (in.)    =  1  foot  (ft.). 

3  feet  =  1  yard  (yd.). 
5i  yards  or  16^  feet  =  1  rod  (rd.). 

320  rods  =  1  mile  (mi.). 


SURFACE  MEASURE 

144  square  inches  (sq.  in.)  =  1  square  foot  (sq.  ft.). 

9  square  feet  =  1  square  yard  (sq.  yd.). 

30|  square  yards  =  1  square  rod  (sq.  rd.). 

160  square  rods  =  1  acre  (A.). 

640  acres  =  1  square  mile  (sq.  mi.). 

262 


TABLES  263 


VOLUME  MEASURE 

1728  cubic  inches  (cu.  in.)  =  1  cubic  foot  (cu.  ft.). 
27  cubic  feet  =  1  cubic  yard  (cu.  yd.). 

TIME 

60  seconds  (sec.)  =  1  minute  (nain.). 
60  minutes  =  1  hour  (hr.). 

24  hours  =  1  day  (da.). 

7  days  =  1  week  (wk.). 

365  days  =  1  common  year  (yr.), 

366  days  =  1  leap  year. 

COUNTING 

12  =1  dozen  (doz.). 

12  dozen  =  1  gross. 
20  =1  score. 

PAPER  MEASURE 

24  sheets  =  1  quire. 
20  quires  =  1  ream. 

ARC  AND  ANGLE  MEASURE 

60  seconds  (")  =  1  minute  ('). 
60  minutes  =  1  degree  (°). 
An  arc  of  360°  =  1  circumference. 

UNITED  STATES  MONEY 

10  mills    =  1  cent. 
10  cents   =  1  dime. 
10  dimes  =  1  dollar. 

TROY  WEIGHT 

24  grains  (gr.)     =1  pennyweight  (pwt.). 
20  pennyweights  =  1  ounce  (oz.). 
12  ounces  =  1  pound  (lb.). 


254  TABLES 

APOTHECARIES »  WEIGHT 

20  grains  (gr.)     =  1  scruple  (sc.  or  3). 
3  scruples  =  1  dram  (dr.  or  3). 

8  drams  =  1  ounce  (oz.  or  3  ). 

EQUIVALENTS 

1  gallon  =  231  cubic  inches. 

1  bushel  =  2150.42  cubic  inches. 

1  quart,  dry  measure  =  1|  quarts,  liquid  measure  (nearly), 

1  lb.  Avoirdupois        =  7000  gr. 

1  oz.  Avoirdupois       =  437.5  gr. 

1  lb.  Troy  or  Apoth.  =  5760  gr. 

1  oz.  Troy  or  Apoth.  =  480  gr. 

1  lb.  Avoirdupois       =  lyV?  lb.  Troy  or  Apoth. 

1  oz.  Avoirdupois       =  ^|f  oz.  Troy  or  Apoth. 


THE  MULTIPLICATION  TABLE 


255 


2x    1=   2 

3x    1=   3 

4x    1=    4 

5x    1=    5 

2x    2=    4 

3x    2=    6 

4x    2=    8 

5x    2  =  10 

2x    3=    6 

3x3=9 

4x    3  =  12 

5x    3  =  15 

2x    4=    8 

3x    4  =  12 

4  X    4  =  16 

5x    4  =  20 

2x    5  =  10 

3x    5  =  15 

4  X    5  =  20 

5x    5  =  25 

2x    6  =  12 

3x    6  =  18 

4x    6  =  24 

5  X    6  =  30 

2x    7  =  14 

3  x    7  =  21 

4x    7  =  28 

5x    7  =  35 

2x    8  =  16 

3x    8  =  24 

4  X    8  =  32 

5x    8  =  40 

2  x    9  =  18 

3  X    9  =  27 

4x    9  =  36 

5x    9  =  45 

2  X  10  =  20 

3  X  10  =  30 

4  X  10  =  40 

5  X  10  =  50 

2  X  11  =  22 

3  X  11  =  33 

4  x  11  =  44 

5  X  11  =  55 

2  X  12  =  24 

3  X  12  =  36 

4  X  12  =  48 

5  X  12  =  60 

6X    1=    6 

7x    1=    7 

8x    1=    8 

9x    1=     9 

6x    2  =  12 

7X    2  =  14 

8x    2  =  16 

9x    2=    18 

6x    3  =  18 

7x    3  =  21 

8x    3  =  24 

9x    3=   27 

6x    4  =  24 

7x    4  =  28 

8x    4  =  32 

9x    4=   36 

6  X    5  =  30 

7x    5  =  35 

8x    5  =  40 

9x    5=   45 

6  X    6  =  36 

7x    6  =  42 

8x    6  =  48 

9x    6=    54 

6x    7  =  42 

7  X    7  =  49 

8x    7  =  56 

9x    7=    63 

6x    8  =  48 

7x    8  =  56 

8x    8  =  64 

9x    8=    72 

6  X    9  =  54 

7x    9  =  63 

8x    9  =  72 

9x    9=    81 

6  X  10  =  60 

7  X  10  =  70 

8  X  10  =  80 

9  X  10  =    90 

6  X  11  =  66 

7  X  11  =  77 

8  X  11  =  88 

9x11=    99 

6  X  12  =  72 

7  X  12  =  84 

8  X  12  =  96 

9  X  12  =  108 

10  X    1  =    10 

11  X    1=    11 

12  X    1=    12 

KOMAN 

10  X    2=    20 

11  X    2  =    22 

12  X    2=    24 

Numerals 

10  X    3=   30 

11  X    3=    33 

12  X    3  =    36 

I  =1 

10  X    4=   40 

11  X    4  =   44 

12  X    4=    48 

10  X    5=   50 

11  X    5  =    55  ■ 

12  X    5  =    60 

V=5 

10  X    6  =    60 

11  X    6  =    66 

12  X    6  =    72 

X=10 

10  X    7=    70 

11  X    7  =   77 

12  X    7=    84 

L  =50 

10  X    8=    80 

11  X    8=   88 

12  X    8  =    96 

C  =100 

10  X    9=    90 

11  X    9  =   99 

12  X    9  =  108 

D=500 

10  X  10  =  100 

11  X  10  =  110 

12  X  10  =  120 

10  X  11  =  110 

11  X  11  =  121 

12  V  11  =  132 

M  =  1000 

10  X  12  =  120 

11  X  12  =  132 

12  X  12  =  144 

M  =  1,000,000 

INDEX 


Accounts  and  bills,  117-121. 
Addend,  7. 
Addition,  7. 

of  compound  numbers,  190. 

of  decimals,  95. 

of  fractions  and  mixed  numbers,  61-62. 

of  integers,  7-10. 

terms  of,  7. 
Aliquot  parts,  86-87. 
Altitude,  205. 
Angles,  177-180. 

acute,  179. 

around  a  point,  178. 

obtuse,  179. 

right,  179. 

sides  of,  178. 
Apothecaries'  weight,  182. 
Arc  and  angle  measure,  177-180. 
Areas  of  parallelograms,  205-206. 
Areas  of  triangles,  207-208, 
Articles  sold  by  the  100,  1000,  and  cwt., 

203. 
Averages,  79,  80. 
Avoirdupois  weight,  168. 

Balance  of  an  account,  118. 
Bale,  176. 
Barrel,  166. 
Base  of  a  figure,  204. 
Bills,  117-121. 
Board  foot,  219. 
Building  walls,  211. 
Bundle,  176. 

Cancellation,  39. 
Century,  174. 
Circle,  180. 
Circumference,  180. 
Coins,  181. 
Compound  number,  166. 


Contents  or  volume,  122. 
Cord,  209-210. 
Counting,  175. 
Credit,  118. 
Creditor,  118. 
Cube,  121. 
Cubic  foot,  122. 
Cubic  inch,  121. 

pattern  of,  123. 
Cubic  inches  in  a  gallon,  126,  167. 
Cubic  yard,  122. 

Days  in  a  month,  week,  and  year,  174. 

Debtor,  118. 

Decade,  174. 

Decimal  point,  89. 

Decimals,  88-106, 

Degrees,  177-180. 

Denominate  numbers,  166-193. 

Denomination,  166. 

Denominator,  51. 

common,  56. 

least  common,  56. 
Dimensions  of  a  solid,  121. 
Dividend,  21,  51. 
Division,  21. 

by  multiples  of  ten,  97. 

by  twenty-five,  116. 

fraction  an  expression  of,  51. 

of  decimals,  100. 

of  fractions,  75. 

of  integers,  21-24. 

terms  of,  21. 

Exact  differences  between  dates,  193. 

Factors,  17,  21,  36. 

integral,  36. 

prime,  36. 
Figures,  3. 


266 


INDEX 


257 


Figures,  significant,  3. 
Floor  covering,  212-215. 
Fluid  ounce,  183. 
Fractions,  51. 

at  the  end  of  a  decimal,  106. 

clearing  divisor  of,  157. 

common,  91. 

complex,  77. 

compound,  68. 

improper,  53. 

in  the  multiplicand,  156. 

proper,  53. 

simple,  77. 

terms  of,  51. 

value  of,  51. 

which  can  be  reduced  to  exact  deci- 
mals, 105. 

Greatest  common  divisor,  44-45. 

Hogshead,  166. 

Ideas  of  proportion,  34,  35,  81. 
Indicated  work,  30,  31. 
Integer,  36. 
Integral  factor,  36. 
Interest,  231-243. 

Least  common  multiple,  42-44. 
Legal  rate  of  interest,  232. 
Linear  measure,  169-170. 
Lumber  measure,  219-222. 

Measurements,  204-222. 
Minuend,  11. 
Mixed  decimal,  91. 
Multiple,  36. 

common,  42. 

least  common,  42. 
Multiplicand,  17 
Multiplication,  17. 

by  multiples  of  ten,  97. 

by  twenty-five,  116. 

of  compound  numbers,  194. 

of  decimals,  99. 

of  fractions,  68-73. 

of   fractions    illustrated   graphically, 
70-71. 

of  integers,  17-20. 

terms  of,  17. 
Multiplication  and  division  combined,  67. 


Multiplier,  17. 

Naught,  3. 
Notation,  3. 

Arabic,  3,  4. 

Roman,  3,  6. 
Number,  3. 

Numbers  prime  to  each  other,  44. 
Numeration,  5. 
Numerator,  51. 

Of  between  fractions,  68. 

Paper  measure,  176. 

Parallel  lines,  204. 

Paral'elogram,  204. 

Parenthesis, -30. 

Parties  to  an  account,  118. 

Per  cent,  140. 

Percentage,  140-145,  151-156. 

Percents  equivalent  to  common  fractions, 

150. 
Perimeter,  10. 
Periods,  4. 

Perpendicular  lines,  204. 
Places,  4. 
Plas    ring,  216. 
Power,  90. 
Principal,  231. 
Principles 

of  Arabic  notation,  32. 

of  Roman  notation,  6. 
Product,  17. 
Products  and  factors,  133,  137-139. 

Quadrilateral,  204. 
Quick  test,  67,  129. 
Quotient,  21. 

Rate  of  interest,  232. 
Receipt  of  bill,  119. 
Rectangle,  204. 
Rectangular  prism,  121. 
Reduction,  51. 

ascending,  183,  187. 

descending,  183,  184. 

of  common  fractions  and  mixed  num- 
bers to  decimals,  104. 

of  decimals  to  common  fractions  or 
mixed  numbers,  103. 


258 


INDEX 


Reduction 

of  denominate  numbers,  18^189. 
of  fractions  to  integers  or  mixed  num- 
bers, 53. 
of  fractions  to  least  common  denomi- 
nator, 57. 
of  fractions  to  lowest  terms,  52. 
Remainder,  11,  21,  23. 
Review  and  practice,  13-16,  25-29,  40-41, 
58-60,  74,   82-85,   107-115,    129-132, 
146-149,   158-165,    195-202,   223-230, 
244-251. 
Review  of  fractions  studied  in  primary 

arithmetic,  49-50, 
Review  of  integers,  46-48. 
Review  of  primary  arithmetic,  17-18. 
Rule 

for  finding  the  number  of  board  feet  in 

a  piece  of  lumber,  220. 
for  finding  whether  a  number  is  prime 

or  composite,  37. 
for  placing  the  decimal  point,  100. 

Separatrix,  4. 

Simplest  form,  61. 

Solid,  121. 

Special  cases  in  multiplication  and  divi- 
sion, 32-33. 

Statements  and  questions  of  relation,  134- 
186. 


Subtraction,  11. 

of  compound  numbers,  191. 

of  decimals,  95. 

of  fractions  and  mixed  numbers,  63-65, 

of  integers,  11-13. 

terms  of,  11. 
Subtrahend,  11. 
Sum,  7. 
Surface  measure,  171. 

Terms  of  a  fraction,  51. 
Test 

of  addition,  9. 

of  division,  24. 

of  multiplication,  33. 

of  subtraction,  12. 
Time,  174. 
Ton,  168. 
Triangle,  207. 
Troy  weight,  182. 

Unit,  3. 

United  States  money,  181. 

Usury,  232. 

Volume  measure,  121,  128,  173. 

Wall  coverings,  217,  218. 

Zero.  3. 


H6 


^305^83 

U/i  163 

vy 


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